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Transcript of Luca Bai Otti
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 1100
Luca Baiotti
Osaka University Institute for Laser Engineering
Introduction to numerical relativity
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 2100
bull Overview of general relativity
bull Gravitational waves
bull Gravitational-wave detection
bull Numerical relativity bull 3+1 decomposition
bull Einstein equations
bull gauge conditions
bull gravitational-wave extraction
bull non-vacuum spacetimesbull general-relativistic magnetohydrodynamics
bull Some applications binary neutron-star merger
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 3100
bull General Relativity is Einsteinrsquos theory of gravity (1915)
bull At the beginning of the XX century there were no observational inconsistencies with
Newtonrsquos theory of gravitation
bull But Newtonian gravity is inconsistent with the principle of causality of special
relativity nothing can move faster than the speed of light
bull Indeed Newtonrsquos inverse square law implies action at a distance If one object moves
the other one knows about it instantaneously due to the change in the gravitational force
no matter how big their separation is
bull The work of Einstein to make gravity consistent with special relativity brought to a
major revision of how we think of space and time It started with a simple principle now
known as the equivalence principle
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 4100
bull The equivalence principle in few common words ldquoAll things fall in the same wayrdquo
bull Or slightly more wordy ldquoall objects have the same acceleration in a gravitational
fieldrdquo (eg a feather and bowling ball fall with the same acceleration in the absence of
air friction)
bull The fact that ldquoAll things fall in the same wayrdquo is true because the ldquoinertialrdquo mass
that enters Newtonrsquos law of motion F=ma is the same as the ldquogravitationalrdquo mass
that enters the gravitational-force law The principle of equivalence is really a
statement that inertial and gravitational masses are equal to each other for anyobject
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 5100
bull Einstein reasoned that a uniform gravitational field in some
direction is indistinguishable from a uniform acceleration in
the opposite direction
bull And so postulated his version of the equivalence principlewhich puts gravity and acceleration on an equal footing
bull ldquowe [] assume the complete physical equivalence of a
gravitational field and a corresponding acceleration of
the reference system
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 6100
bull Einstein started to think of the path of an object as a property of spacetime itself rather than
being related with the specific properties of the object
bull The idea is that gravity is a manifestation of the fact that objects in free fall follow
geodesics in curved spacetimes
bull What are geodesics
bull We know in our ordinary experience (flat spacetime) that in the absence of any forces
objects follow straight lines and we also know that straight lines are the shortest possible
paths that connect two points in such conditions
bull The generalization of the notion of a ldquostraight linerdquo valid also in curved spacetimes is called
geodesic
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 7100
bull A famous story to simply illustrate the idea of general relativity is the Parable of the Apple
by Misner Thorne and Wheeler [Gravitation (1973)]
bull The parable tries to explain the nature of gravitation in terms of the curvature of spacetime
The spacetime of the parable is the two-dimensional curved surface of an apple
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 8100
bull The tale goes like this One day a student reflecting on the difference between Einsteins and
Newtons views about gravity noticed ants running on the surface of an apple
bull By advancing alternately and of the same amount the left and right legs the ants seemed to take
the most economical path ldquowow they are going along geodesics on this surfacerdquo
bull The student followed the path of an ant tracing it and then cutting with a knife a small stripe
around the trace ldquoIndeed when put on a plane the path is a straight linerdquo
bull Each geodesic may be regarded as a path (world line) of a free particle on this surface (taken as
a two-dimensional spacetime)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 9100
bull Then the student looked at two ants going from the same spot onto initially divergent paths but
then when approaching the top (near the dimple) of the apple the paths crossed and continued
into different directions
bull The reason of the curved trajectories is
bull According to Newton gravitation is acting at a distance from a center of attraction (the
dimple)
bull According to Einstein the local geometry of the surface around the dimple is curved
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 10100
bull Comments
bull Einstein interpretation dispenses with any action-at-a-distance
bull Although the surface of the apple is curved if you look at any local spot closely (with amagnifying glass) its geometry looks like that of a flat surface (the Minkowski spacetime of
the special relativity)
bull The interaction of spacetime and matter is summarized in John Archibald Wheelers favorite
words spacetime tells matter how to move and matter tells spacetime how to curve
bull This reciprocal influence (matter spacetime) makes Einsteinrsquos field equation non-linearand so very hard to solve
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 11100
bull Summary of the parable
bull 1) objects follow geodesics and locally geodesics appear straight
bull 2) over more extended regions of space and time geodesics originally receding from eachother begin to approach at a rate governed by the curvature of spacetime and this effect of
geometry on matter is what was called ldquogravitationrdquo
bull 3) matter in turn warps geometry
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 12100
bull Important quantities in general relativity
bull the metric (the metric tensor g) which may be regarded as a machinery for measuring
distances
dS 2 = gmicroν dxmicrodxν
Rmicroν = Rα
microαν bull the Ricci tensor and the curvature (Ricci) scalar R = gmicroν Rmicroν
Rαβmicroν = Γ
αβν micro minus Γ
αβmicroν + Γ
ασmicroΓ
σβν minus Γ
ασν Γ
σβmicro
Γαβmicro =
12
gασ(gβσmicro + gσmicroβ minus gβmicroσ)
bull curvature expressed by the Riemann curvature tensor
(where are the Christoffel symbols)
bull covariant derivative a derivative that takes into account the curvature of the spacetime
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 13100
bull From these quantities the path of any particle can be calculated This is how geometry tells
matter how to move
bull The other direction (matter tells spacetime how to curve) requires to know the distribution of
matter (massenergymomentum) described through the stress-energy tensor T
bull After many years of thinking Einstein reached a satisfactory form for the equations relating
geometry and matter
Einstein_tensor = constant x T
bull The Einstein tensor (usually called G) is a tensor in 4D spacetime that has the wanted properties
of
bull being a symmetric tensor (it must because the stress-energy tensor is symmetric)
bull having vanishing (covariant) divergence (it must because the stress-energy tensor has
vanishing divergence)
bull the weak-field limit of the Einstein equations gives the Newtonian Poisson equation (from the
comparison to which the value of the above constant is found)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 14100
ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
The Einstein equations
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 15100
bull Gravitational redshift
bull change in the frequency of light moving in regions of different curvature
bull measured in experiments and taken into account by the GPS
bull Periastron shift
bull first measured in the perihelion advance of Mercury (the GR prediction coincides with
the ldquoanomalousrdquo advance if computed in Newtonian theory)bull binary pulsar (strong fields so larger shifts)
bull Bending of light
bull first measured in the bending of photons traveling near the Sun
bull gravitational lensing
bull
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 16100
bull A black hole is literally a region from which light cannot escape
bull The boundary of a black hole is called the event horizon When something passes through the
event horizon it can no longer communicate in any way with the world outside However it is
not a concrete surface there is no local measurement that can tell an observer that heshe is on
the event horizon
bull General relativity predicts that black holes can form by gravitational collapse Once a star has
burned up its supply of nuclear fuel it can no longer support itself against its own weight and it
collapses It will reach a critical density at which an explosive re-ignition of burning occurs
called a supernova
bull There are several candidate objects that are thought to be black holes (because they are very
compact) but there has been no direct observation of black holes up to now
bull The definitive way to detect black holes is through gravitational waves (more on this later)
because gravitational waves from black holes are distinct from those of other objects
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 17100
bull In general relativity disturbances in the spacetime curvature (the ldquooldrdquo ldquogravitational
fieldrdquo) propagate at the speed of light as gravitational radiation or gravitational waves
(also known as gravity waves but this term was already in use in fluid dynamics with a
different meaning so I recommend to avoid it)
bull Gravitational waves are a strong point of general relativity which solves the action-at-a-
distance problem of Newtonian gravity
bull Analogously to light which is produced by the motion of electric charges gravitational
waves are produced by the motion of anything (mass-energy) When you wave yourhand you make gravitational waves
bull The point is that the amplitude of gravitational radiation is very small
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 18100
T R3(GM )
part 3Qij
part t3
MR2
T 3
Mv2
T
LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
LGW = 15Gc5
part 3
Qij
part t3part 3
Qij
part t3 1048575
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 19100
LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
G4
c5
82times
10
minus74
cgs units
c5 27 times 10minus60 cgs units
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 20100
bull In fact gravitational waves have not been detected yet even if there are strong
indications that gravitational waves exist as predicted by general relativity
bull Such an indirect evidence comes from binary pulsars Two neutron stars orbiting each
other at least one of which is a pulsar The curvature is large so it is an interesting place
to study general relativistic effects (Incidentally their orbit may precess by 4 degree per
year)bull The spin axis of pulsars is not aligned with
their magnetic axis so at each rotation theyemit pulses of radio waves and we can
detect them from the Earth Pulsars behave
as very accurate clocks
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 21100
bull Actually such measurements are very precise and
confirm that the orbit is shrinking with time at a rate
in complete agreement with the emission of
gravitational waves predicted by general relativity
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 22100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 23100
radio
far-IR
mid-IR
near-IR
opticalx-ray
gamma-ray
GWs
It has happened over and over in the history of astronomy as a newldquowindowrdquo has been opened a ldquonewrdquo universe has been revealed
The same will happen with GW-astronomy
GSFCNASA
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 24100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 25100
Large-scale
Cryogenic
Gravitational-waveTelescope
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 26100
o Binary neutron stars
o Binary black-holes
o Deformed compact stars including Low Mass X-ray Binaries (LMXB)and pulsars
o Cosmological stochastic background
o Mixed binary systems
o Gravitational collapse (supernovae neutron stars)
o Extreme-Mass-Ratio Inspirals (EMRI)
o
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 27100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 28100
TAMA-Tokyo GEO-Hannover LIGO-Livingston LIGO-Hanford VIRGO-Cascina
and LCGT will measure LL ~ h lt 10-21 with SN~1
Numericalrelativity
h
Knowledge of thewaveforms cancompensate for the
very small SN(matched-filtering) and
so enhance detection and allow for source-characterization
possible
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 29100
ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
The fundamental equations standard numerical relativity aims at solving
are the Einstein equations
8182019 Luca Bai Otti
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General relativity states that our World is a 4D and curvedspacetime and the Einstein equations describe its dynamics
How to solve the Einstein equations numerically
Prominently there is no a priori concept of ldquoflowing of timerdquo (we arenot involved in thermodynamics here) time is just one of thedimensions and on the same level as space dimensions
There is a successful recipe though
8182019 Luca Bai Otti
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We have the illusion to live in 3D and it is easier to tell computers toperform simulations (time-)step by (time-)step
Also assign a normalizationsuch that
Define therefore
So given a manifold describing a spacetime with 4-metricwe want to foliate it via space-like three-dimensionalhypersurfaces We label such hypersurfaces with the
time coordinate t
(ldquothe direction of timerdquo)
(As mostly in numer ical relativity the signature is here -+++)
The function is called the rdquolapserdquo function and itis strictly positive for spacelike hypersurfaces
α
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so that
ii) and the spatial metric
Letrsquos also define
i) the unit normal vector to the hypersurface
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The spatial part is obtained by contracting with the spatialprojection operator defined as
By using we can decompose any 4D tensor into a
purely spatial part (hence in ) and a purely timelike part (henceorthogonal to and aligned with )
while the timelike part is obtained by contracting with the timelikeprojection operator
The two projectors are obviously orthogonal
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The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariantderivative
All the 4D tensors in the Einstein equations can be projectedstraightforwardly onto the 3D spatial slice
In particular the 3D connection coefficients
the 3D Riemann tensor
and the 3D contractions of the 3D Riemann tensor ie the 3DRicci tensor the 3D Ricci scalar and
8182019 Luca Bai Otti
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Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic
curvatureThe extrinsic curvature is defined in terms of the unit normal to as
where is the Lie derivative along
This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric
We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic
curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold
and this can be shown to be equivalent also to
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 36100
Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
8182019 Luca Bai Otti
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
8182019 Luca Bai Otti
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Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 39100
Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
8182019 Luca Bai Otti
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 41100
Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
8182019 Luca Bai Otti
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 43100
Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
8182019 Luca Bai Otti
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 45100
8182019 Luca Bai Otti
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 47100
In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
8182019 Luca Bai Otti
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 49100
Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 50100
We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 51100
We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 52100
First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
8182019 Luca Bai Otti
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 54100
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 55100
Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
8182019 Luca Bai Otti
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
8182019 Luca Bai Otti
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
8182019 Luca Bai Otti
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
8182019 Luca Bai Otti
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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httpslidepdfcomreaderfullluca-bai-otti 62100
The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 65100
Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 68100
Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 73100
In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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8182019 Luca Bai Otti
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
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W f l i EOS
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
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bull Overview of general relativity
bull Gravitational waves
bull Gravitational-wave detection
bull Numerical relativity bull 3+1 decomposition
bull Einstein equations
bull gauge conditions
bull gravitational-wave extraction
bull non-vacuum spacetimesbull general-relativistic magnetohydrodynamics
bull Some applications binary neutron-star merger
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bull General Relativity is Einsteinrsquos theory of gravity (1915)
bull At the beginning of the XX century there were no observational inconsistencies with
Newtonrsquos theory of gravitation
bull But Newtonian gravity is inconsistent with the principle of causality of special
relativity nothing can move faster than the speed of light
bull Indeed Newtonrsquos inverse square law implies action at a distance If one object moves
the other one knows about it instantaneously due to the change in the gravitational force
no matter how big their separation is
bull The work of Einstein to make gravity consistent with special relativity brought to a
major revision of how we think of space and time It started with a simple principle now
known as the equivalence principle
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bull The equivalence principle in few common words ldquoAll things fall in the same wayrdquo
bull Or slightly more wordy ldquoall objects have the same acceleration in a gravitational
fieldrdquo (eg a feather and bowling ball fall with the same acceleration in the absence of
air friction)
bull The fact that ldquoAll things fall in the same wayrdquo is true because the ldquoinertialrdquo mass
that enters Newtonrsquos law of motion F=ma is the same as the ldquogravitationalrdquo mass
that enters the gravitational-force law The principle of equivalence is really a
statement that inertial and gravitational masses are equal to each other for anyobject
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bull Einstein reasoned that a uniform gravitational field in some
direction is indistinguishable from a uniform acceleration in
the opposite direction
bull And so postulated his version of the equivalence principlewhich puts gravity and acceleration on an equal footing
bull ldquowe [] assume the complete physical equivalence of a
gravitational field and a corresponding acceleration of
the reference system
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bull Einstein started to think of the path of an object as a property of spacetime itself rather than
being related with the specific properties of the object
bull The idea is that gravity is a manifestation of the fact that objects in free fall follow
geodesics in curved spacetimes
bull What are geodesics
bull We know in our ordinary experience (flat spacetime) that in the absence of any forces
objects follow straight lines and we also know that straight lines are the shortest possible
paths that connect two points in such conditions
bull The generalization of the notion of a ldquostraight linerdquo valid also in curved spacetimes is called
geodesic
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bull A famous story to simply illustrate the idea of general relativity is the Parable of the Apple
by Misner Thorne and Wheeler [Gravitation (1973)]
bull The parable tries to explain the nature of gravitation in terms of the curvature of spacetime
The spacetime of the parable is the two-dimensional curved surface of an apple
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bull The tale goes like this One day a student reflecting on the difference between Einsteins and
Newtons views about gravity noticed ants running on the surface of an apple
bull By advancing alternately and of the same amount the left and right legs the ants seemed to take
the most economical path ldquowow they are going along geodesics on this surfacerdquo
bull The student followed the path of an ant tracing it and then cutting with a knife a small stripe
around the trace ldquoIndeed when put on a plane the path is a straight linerdquo
bull Each geodesic may be regarded as a path (world line) of a free particle on this surface (taken as
a two-dimensional spacetime)
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bull Then the student looked at two ants going from the same spot onto initially divergent paths but
then when approaching the top (near the dimple) of the apple the paths crossed and continued
into different directions
bull The reason of the curved trajectories is
bull According to Newton gravitation is acting at a distance from a center of attraction (the
dimple)
bull According to Einstein the local geometry of the surface around the dimple is curved
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bull Comments
bull Einstein interpretation dispenses with any action-at-a-distance
bull Although the surface of the apple is curved if you look at any local spot closely (with amagnifying glass) its geometry looks like that of a flat surface (the Minkowski spacetime of
the special relativity)
bull The interaction of spacetime and matter is summarized in John Archibald Wheelers favorite
words spacetime tells matter how to move and matter tells spacetime how to curve
bull This reciprocal influence (matter spacetime) makes Einsteinrsquos field equation non-linearand so very hard to solve
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bull Summary of the parable
bull 1) objects follow geodesics and locally geodesics appear straight
bull 2) over more extended regions of space and time geodesics originally receding from eachother begin to approach at a rate governed by the curvature of spacetime and this effect of
geometry on matter is what was called ldquogravitationrdquo
bull 3) matter in turn warps geometry
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bull Important quantities in general relativity
bull the metric (the metric tensor g) which may be regarded as a machinery for measuring
distances
dS 2 = gmicroν dxmicrodxν
Rmicroν = Rα
microαν bull the Ricci tensor and the curvature (Ricci) scalar R = gmicroν Rmicroν
Rαβmicroν = Γ
αβν micro minus Γ
αβmicroν + Γ
ασmicroΓ
σβν minus Γ
ασν Γ
σβmicro
Γαβmicro =
12
gασ(gβσmicro + gσmicroβ minus gβmicroσ)
bull curvature expressed by the Riemann curvature tensor
(where are the Christoffel symbols)
bull covariant derivative a derivative that takes into account the curvature of the spacetime
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bull From these quantities the path of any particle can be calculated This is how geometry tells
matter how to move
bull The other direction (matter tells spacetime how to curve) requires to know the distribution of
matter (massenergymomentum) described through the stress-energy tensor T
bull After many years of thinking Einstein reached a satisfactory form for the equations relating
geometry and matter
Einstein_tensor = constant x T
bull The Einstein tensor (usually called G) is a tensor in 4D spacetime that has the wanted properties
of
bull being a symmetric tensor (it must because the stress-energy tensor is symmetric)
bull having vanishing (covariant) divergence (it must because the stress-energy tensor has
vanishing divergence)
bull the weak-field limit of the Einstein equations gives the Newtonian Poisson equation (from the
comparison to which the value of the above constant is found)
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ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
The Einstein equations
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
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bull Gravitational redshift
bull change in the frequency of light moving in regions of different curvature
bull measured in experiments and taken into account by the GPS
bull Periastron shift
bull first measured in the perihelion advance of Mercury (the GR prediction coincides with
the ldquoanomalousrdquo advance if computed in Newtonian theory)bull binary pulsar (strong fields so larger shifts)
bull Bending of light
bull first measured in the bending of photons traveling near the Sun
bull gravitational lensing
bull
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bull A black hole is literally a region from which light cannot escape
bull The boundary of a black hole is called the event horizon When something passes through the
event horizon it can no longer communicate in any way with the world outside However it is
not a concrete surface there is no local measurement that can tell an observer that heshe is on
the event horizon
bull General relativity predicts that black holes can form by gravitational collapse Once a star has
burned up its supply of nuclear fuel it can no longer support itself against its own weight and it
collapses It will reach a critical density at which an explosive re-ignition of burning occurs
called a supernova
bull There are several candidate objects that are thought to be black holes (because they are very
compact) but there has been no direct observation of black holes up to now
bull The definitive way to detect black holes is through gravitational waves (more on this later)
because gravitational waves from black holes are distinct from those of other objects
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bull In general relativity disturbances in the spacetime curvature (the ldquooldrdquo ldquogravitational
fieldrdquo) propagate at the speed of light as gravitational radiation or gravitational waves
(also known as gravity waves but this term was already in use in fluid dynamics with a
different meaning so I recommend to avoid it)
bull Gravitational waves are a strong point of general relativity which solves the action-at-a-
distance problem of Newtonian gravity
bull Analogously to light which is produced by the motion of electric charges gravitational
waves are produced by the motion of anything (mass-energy) When you wave yourhand you make gravitational waves
bull The point is that the amplitude of gravitational radiation is very small
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T R3(GM )
part 3Qij
part t3
MR2
T 3
Mv2
T
LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
LGW = 15Gc5
part 3
Qij
part t3part 3
Qij
part t3 1048575
8182019 Luca Bai Otti
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LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
G4
c5
82times
10
minus74
cgs units
c5 27 times 10minus60 cgs units
8182019 Luca Bai Otti
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bull In fact gravitational waves have not been detected yet even if there are strong
indications that gravitational waves exist as predicted by general relativity
bull Such an indirect evidence comes from binary pulsars Two neutron stars orbiting each
other at least one of which is a pulsar The curvature is large so it is an interesting place
to study general relativistic effects (Incidentally their orbit may precess by 4 degree per
year)bull The spin axis of pulsars is not aligned with
their magnetic axis so at each rotation theyemit pulses of radio waves and we can
detect them from the Earth Pulsars behave
as very accurate clocks
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bull Actually such measurements are very precise and
confirm that the orbit is shrinking with time at a rate
in complete agreement with the emission of
gravitational waves predicted by general relativity
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radio
far-IR
mid-IR
near-IR
opticalx-ray
gamma-ray
GWs
It has happened over and over in the history of astronomy as a newldquowindowrdquo has been opened a ldquonewrdquo universe has been revealed
The same will happen with GW-astronomy
GSFCNASA
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Large-scale
Cryogenic
Gravitational-waveTelescope
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o Binary neutron stars
o Binary black-holes
o Deformed compact stars including Low Mass X-ray Binaries (LMXB)and pulsars
o Cosmological stochastic background
o Mixed binary systems
o Gravitational collapse (supernovae neutron stars)
o Extreme-Mass-Ratio Inspirals (EMRI)
o
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TAMA-Tokyo GEO-Hannover LIGO-Livingston LIGO-Hanford VIRGO-Cascina
and LCGT will measure LL ~ h lt 10-21 with SN~1
Numericalrelativity
h
Knowledge of thewaveforms cancompensate for the
very small SN(matched-filtering) and
so enhance detection and allow for source-characterization
possible
8182019 Luca Bai Otti
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ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
The fundamental equations standard numerical relativity aims at solving
are the Einstein equations
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General relativity states that our World is a 4D and curvedspacetime and the Einstein equations describe its dynamics
How to solve the Einstein equations numerically
Prominently there is no a priori concept of ldquoflowing of timerdquo (we arenot involved in thermodynamics here) time is just one of thedimensions and on the same level as space dimensions
There is a successful recipe though
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 31100
We have the illusion to live in 3D and it is easier to tell computers toperform simulations (time-)step by (time-)step
Also assign a normalizationsuch that
Define therefore
So given a manifold describing a spacetime with 4-metricwe want to foliate it via space-like three-dimensionalhypersurfaces We label such hypersurfaces with the
time coordinate t
(ldquothe direction of timerdquo)
(As mostly in numer ical relativity the signature is here -+++)
The function is called the rdquolapserdquo function and itis strictly positive for spacelike hypersurfaces
α
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so that
ii) and the spatial metric
Letrsquos also define
i) the unit normal vector to the hypersurface
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The spatial part is obtained by contracting with the spatialprojection operator defined as
By using we can decompose any 4D tensor into a
purely spatial part (hence in ) and a purely timelike part (henceorthogonal to and aligned with )
while the timelike part is obtained by contracting with the timelikeprojection operator
The two projectors are obviously orthogonal
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The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariantderivative
All the 4D tensors in the Einstein equations can be projectedstraightforwardly onto the 3D spatial slice
In particular the 3D connection coefficients
the 3D Riemann tensor
and the 3D contractions of the 3D Riemann tensor ie the 3DRicci tensor the 3D Ricci scalar and
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Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic
curvatureThe extrinsic curvature is defined in terms of the unit normal to as
where is the Lie derivative along
This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric
We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic
curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold
and this can be shown to be equivalent also to
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 36100
Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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httpslidepdfcomreaderfullluca-bai-otti 38100
Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
8182019 Luca Bai Otti
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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httpslidepdfcomreaderfullluca-bai-otti 40100
Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
8182019 Luca Bai Otti
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
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Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
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W f l i EOS
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 3100
bull General Relativity is Einsteinrsquos theory of gravity (1915)
bull At the beginning of the XX century there were no observational inconsistencies with
Newtonrsquos theory of gravitation
bull But Newtonian gravity is inconsistent with the principle of causality of special
relativity nothing can move faster than the speed of light
bull Indeed Newtonrsquos inverse square law implies action at a distance If one object moves
the other one knows about it instantaneously due to the change in the gravitational force
no matter how big their separation is
bull The work of Einstein to make gravity consistent with special relativity brought to a
major revision of how we think of space and time It started with a simple principle now
known as the equivalence principle
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bull The equivalence principle in few common words ldquoAll things fall in the same wayrdquo
bull Or slightly more wordy ldquoall objects have the same acceleration in a gravitational
fieldrdquo (eg a feather and bowling ball fall with the same acceleration in the absence of
air friction)
bull The fact that ldquoAll things fall in the same wayrdquo is true because the ldquoinertialrdquo mass
that enters Newtonrsquos law of motion F=ma is the same as the ldquogravitationalrdquo mass
that enters the gravitational-force law The principle of equivalence is really a
statement that inertial and gravitational masses are equal to each other for anyobject
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bull Einstein reasoned that a uniform gravitational field in some
direction is indistinguishable from a uniform acceleration in
the opposite direction
bull And so postulated his version of the equivalence principlewhich puts gravity and acceleration on an equal footing
bull ldquowe [] assume the complete physical equivalence of a
gravitational field and a corresponding acceleration of
the reference system
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bull Einstein started to think of the path of an object as a property of spacetime itself rather than
being related with the specific properties of the object
bull The idea is that gravity is a manifestation of the fact that objects in free fall follow
geodesics in curved spacetimes
bull What are geodesics
bull We know in our ordinary experience (flat spacetime) that in the absence of any forces
objects follow straight lines and we also know that straight lines are the shortest possible
paths that connect two points in such conditions
bull The generalization of the notion of a ldquostraight linerdquo valid also in curved spacetimes is called
geodesic
8182019 Luca Bai Otti
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bull A famous story to simply illustrate the idea of general relativity is the Parable of the Apple
by Misner Thorne and Wheeler [Gravitation (1973)]
bull The parable tries to explain the nature of gravitation in terms of the curvature of spacetime
The spacetime of the parable is the two-dimensional curved surface of an apple
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bull The tale goes like this One day a student reflecting on the difference between Einsteins and
Newtons views about gravity noticed ants running on the surface of an apple
bull By advancing alternately and of the same amount the left and right legs the ants seemed to take
the most economical path ldquowow they are going along geodesics on this surfacerdquo
bull The student followed the path of an ant tracing it and then cutting with a knife a small stripe
around the trace ldquoIndeed when put on a plane the path is a straight linerdquo
bull Each geodesic may be regarded as a path (world line) of a free particle on this surface (taken as
a two-dimensional spacetime)
8182019 Luca Bai Otti
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bull Then the student looked at two ants going from the same spot onto initially divergent paths but
then when approaching the top (near the dimple) of the apple the paths crossed and continued
into different directions
bull The reason of the curved trajectories is
bull According to Newton gravitation is acting at a distance from a center of attraction (the
dimple)
bull According to Einstein the local geometry of the surface around the dimple is curved
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bull Comments
bull Einstein interpretation dispenses with any action-at-a-distance
bull Although the surface of the apple is curved if you look at any local spot closely (with amagnifying glass) its geometry looks like that of a flat surface (the Minkowski spacetime of
the special relativity)
bull The interaction of spacetime and matter is summarized in John Archibald Wheelers favorite
words spacetime tells matter how to move and matter tells spacetime how to curve
bull This reciprocal influence (matter spacetime) makes Einsteinrsquos field equation non-linearand so very hard to solve
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bull Summary of the parable
bull 1) objects follow geodesics and locally geodesics appear straight
bull 2) over more extended regions of space and time geodesics originally receding from eachother begin to approach at a rate governed by the curvature of spacetime and this effect of
geometry on matter is what was called ldquogravitationrdquo
bull 3) matter in turn warps geometry
8182019 Luca Bai Otti
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bull Important quantities in general relativity
bull the metric (the metric tensor g) which may be regarded as a machinery for measuring
distances
dS 2 = gmicroν dxmicrodxν
Rmicroν = Rα
microαν bull the Ricci tensor and the curvature (Ricci) scalar R = gmicroν Rmicroν
Rαβmicroν = Γ
αβν micro minus Γ
αβmicroν + Γ
ασmicroΓ
σβν minus Γ
ασν Γ
σβmicro
Γαβmicro =
12
gασ(gβσmicro + gσmicroβ minus gβmicroσ)
bull curvature expressed by the Riemann curvature tensor
(where are the Christoffel symbols)
bull covariant derivative a derivative that takes into account the curvature of the spacetime
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bull From these quantities the path of any particle can be calculated This is how geometry tells
matter how to move
bull The other direction (matter tells spacetime how to curve) requires to know the distribution of
matter (massenergymomentum) described through the stress-energy tensor T
bull After many years of thinking Einstein reached a satisfactory form for the equations relating
geometry and matter
Einstein_tensor = constant x T
bull The Einstein tensor (usually called G) is a tensor in 4D spacetime that has the wanted properties
of
bull being a symmetric tensor (it must because the stress-energy tensor is symmetric)
bull having vanishing (covariant) divergence (it must because the stress-energy tensor has
vanishing divergence)
bull the weak-field limit of the Einstein equations gives the Newtonian Poisson equation (from the
comparison to which the value of the above constant is found)
8182019 Luca Bai Otti
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ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
The Einstein equations
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
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httpslidepdfcomreaderfullluca-bai-otti 15100
bull Gravitational redshift
bull change in the frequency of light moving in regions of different curvature
bull measured in experiments and taken into account by the GPS
bull Periastron shift
bull first measured in the perihelion advance of Mercury (the GR prediction coincides with
the ldquoanomalousrdquo advance if computed in Newtonian theory)bull binary pulsar (strong fields so larger shifts)
bull Bending of light
bull first measured in the bending of photons traveling near the Sun
bull gravitational lensing
bull
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bull A black hole is literally a region from which light cannot escape
bull The boundary of a black hole is called the event horizon When something passes through the
event horizon it can no longer communicate in any way with the world outside However it is
not a concrete surface there is no local measurement that can tell an observer that heshe is on
the event horizon
bull General relativity predicts that black holes can form by gravitational collapse Once a star has
burned up its supply of nuclear fuel it can no longer support itself against its own weight and it
collapses It will reach a critical density at which an explosive re-ignition of burning occurs
called a supernova
bull There are several candidate objects that are thought to be black holes (because they are very
compact) but there has been no direct observation of black holes up to now
bull The definitive way to detect black holes is through gravitational waves (more on this later)
because gravitational waves from black holes are distinct from those of other objects
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bull In general relativity disturbances in the spacetime curvature (the ldquooldrdquo ldquogravitational
fieldrdquo) propagate at the speed of light as gravitational radiation or gravitational waves
(also known as gravity waves but this term was already in use in fluid dynamics with a
different meaning so I recommend to avoid it)
bull Gravitational waves are a strong point of general relativity which solves the action-at-a-
distance problem of Newtonian gravity
bull Analogously to light which is produced by the motion of electric charges gravitational
waves are produced by the motion of anything (mass-energy) When you wave yourhand you make gravitational waves
bull The point is that the amplitude of gravitational radiation is very small
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T R3(GM )
part 3Qij
part t3
MR2
T 3
Mv2
T
LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
LGW = 15Gc5
part 3
Qij
part t3part 3
Qij
part t3 1048575
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 19100
LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
G4
c5
82times
10
minus74
cgs units
c5 27 times 10minus60 cgs units
8182019 Luca Bai Otti
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bull In fact gravitational waves have not been detected yet even if there are strong
indications that gravitational waves exist as predicted by general relativity
bull Such an indirect evidence comes from binary pulsars Two neutron stars orbiting each
other at least one of which is a pulsar The curvature is large so it is an interesting place
to study general relativistic effects (Incidentally their orbit may precess by 4 degree per
year)bull The spin axis of pulsars is not aligned with
their magnetic axis so at each rotation theyemit pulses of radio waves and we can
detect them from the Earth Pulsars behave
as very accurate clocks
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bull Actually such measurements are very precise and
confirm that the orbit is shrinking with time at a rate
in complete agreement with the emission of
gravitational waves predicted by general relativity
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radio
far-IR
mid-IR
near-IR
opticalx-ray
gamma-ray
GWs
It has happened over and over in the history of astronomy as a newldquowindowrdquo has been opened a ldquonewrdquo universe has been revealed
The same will happen with GW-astronomy
GSFCNASA
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Large-scale
Cryogenic
Gravitational-waveTelescope
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o Binary neutron stars
o Binary black-holes
o Deformed compact stars including Low Mass X-ray Binaries (LMXB)and pulsars
o Cosmological stochastic background
o Mixed binary systems
o Gravitational collapse (supernovae neutron stars)
o Extreme-Mass-Ratio Inspirals (EMRI)
o
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TAMA-Tokyo GEO-Hannover LIGO-Livingston LIGO-Hanford VIRGO-Cascina
and LCGT will measure LL ~ h lt 10-21 with SN~1
Numericalrelativity
h
Knowledge of thewaveforms cancompensate for the
very small SN(matched-filtering) and
so enhance detection and allow for source-characterization
possible
8182019 Luca Bai Otti
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ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
The fundamental equations standard numerical relativity aims at solving
are the Einstein equations
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General relativity states that our World is a 4D and curvedspacetime and the Einstein equations describe its dynamics
How to solve the Einstein equations numerically
Prominently there is no a priori concept of ldquoflowing of timerdquo (we arenot involved in thermodynamics here) time is just one of thedimensions and on the same level as space dimensions
There is a successful recipe though
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We have the illusion to live in 3D and it is easier to tell computers toperform simulations (time-)step by (time-)step
Also assign a normalizationsuch that
Define therefore
So given a manifold describing a spacetime with 4-metricwe want to foliate it via space-like three-dimensionalhypersurfaces We label such hypersurfaces with the
time coordinate t
(ldquothe direction of timerdquo)
(As mostly in numer ical relativity the signature is here -+++)
The function is called the rdquolapserdquo function and itis strictly positive for spacelike hypersurfaces
α
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so that
ii) and the spatial metric
Letrsquos also define
i) the unit normal vector to the hypersurface
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The spatial part is obtained by contracting with the spatialprojection operator defined as
By using we can decompose any 4D tensor into a
purely spatial part (hence in ) and a purely timelike part (henceorthogonal to and aligned with )
while the timelike part is obtained by contracting with the timelikeprojection operator
The two projectors are obviously orthogonal
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The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariantderivative
All the 4D tensors in the Einstein equations can be projectedstraightforwardly onto the 3D spatial slice
In particular the 3D connection coefficients
the 3D Riemann tensor
and the 3D contractions of the 3D Riemann tensor ie the 3DRicci tensor the 3D Ricci scalar and
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Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic
curvatureThe extrinsic curvature is defined in terms of the unit normal to as
where is the Lie derivative along
This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric
We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic
curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold
and this can be shown to be equivalent also to
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Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 61100
There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 62100
The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 63100
Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 64100
We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 65100
Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 66100
where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 67100
conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 68100
Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 72100
Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 73100
In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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8182019 Luca Bai Otti
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
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W f l i EOS
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
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bull The equivalence principle in few common words ldquoAll things fall in the same wayrdquo
bull Or slightly more wordy ldquoall objects have the same acceleration in a gravitational
fieldrdquo (eg a feather and bowling ball fall with the same acceleration in the absence of
air friction)
bull The fact that ldquoAll things fall in the same wayrdquo is true because the ldquoinertialrdquo mass
that enters Newtonrsquos law of motion F=ma is the same as the ldquogravitationalrdquo mass
that enters the gravitational-force law The principle of equivalence is really a
statement that inertial and gravitational masses are equal to each other for anyobject
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bull Einstein reasoned that a uniform gravitational field in some
direction is indistinguishable from a uniform acceleration in
the opposite direction
bull And so postulated his version of the equivalence principlewhich puts gravity and acceleration on an equal footing
bull ldquowe [] assume the complete physical equivalence of a
gravitational field and a corresponding acceleration of
the reference system
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bull Einstein started to think of the path of an object as a property of spacetime itself rather than
being related with the specific properties of the object
bull The idea is that gravity is a manifestation of the fact that objects in free fall follow
geodesics in curved spacetimes
bull What are geodesics
bull We know in our ordinary experience (flat spacetime) that in the absence of any forces
objects follow straight lines and we also know that straight lines are the shortest possible
paths that connect two points in such conditions
bull The generalization of the notion of a ldquostraight linerdquo valid also in curved spacetimes is called
geodesic
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bull A famous story to simply illustrate the idea of general relativity is the Parable of the Apple
by Misner Thorne and Wheeler [Gravitation (1973)]
bull The parable tries to explain the nature of gravitation in terms of the curvature of spacetime
The spacetime of the parable is the two-dimensional curved surface of an apple
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bull The tale goes like this One day a student reflecting on the difference between Einsteins and
Newtons views about gravity noticed ants running on the surface of an apple
bull By advancing alternately and of the same amount the left and right legs the ants seemed to take
the most economical path ldquowow they are going along geodesics on this surfacerdquo
bull The student followed the path of an ant tracing it and then cutting with a knife a small stripe
around the trace ldquoIndeed when put on a plane the path is a straight linerdquo
bull Each geodesic may be regarded as a path (world line) of a free particle on this surface (taken as
a two-dimensional spacetime)
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bull Then the student looked at two ants going from the same spot onto initially divergent paths but
then when approaching the top (near the dimple) of the apple the paths crossed and continued
into different directions
bull The reason of the curved trajectories is
bull According to Newton gravitation is acting at a distance from a center of attraction (the
dimple)
bull According to Einstein the local geometry of the surface around the dimple is curved
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bull Comments
bull Einstein interpretation dispenses with any action-at-a-distance
bull Although the surface of the apple is curved if you look at any local spot closely (with amagnifying glass) its geometry looks like that of a flat surface (the Minkowski spacetime of
the special relativity)
bull The interaction of spacetime and matter is summarized in John Archibald Wheelers favorite
words spacetime tells matter how to move and matter tells spacetime how to curve
bull This reciprocal influence (matter spacetime) makes Einsteinrsquos field equation non-linearand so very hard to solve
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bull Summary of the parable
bull 1) objects follow geodesics and locally geodesics appear straight
bull 2) over more extended regions of space and time geodesics originally receding from eachother begin to approach at a rate governed by the curvature of spacetime and this effect of
geometry on matter is what was called ldquogravitationrdquo
bull 3) matter in turn warps geometry
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bull Important quantities in general relativity
bull the metric (the metric tensor g) which may be regarded as a machinery for measuring
distances
dS 2 = gmicroν dxmicrodxν
Rmicroν = Rα
microαν bull the Ricci tensor and the curvature (Ricci) scalar R = gmicroν Rmicroν
Rαβmicroν = Γ
αβν micro minus Γ
αβmicroν + Γ
ασmicroΓ
σβν minus Γ
ασν Γ
σβmicro
Γαβmicro =
12
gασ(gβσmicro + gσmicroβ minus gβmicroσ)
bull curvature expressed by the Riemann curvature tensor
(where are the Christoffel symbols)
bull covariant derivative a derivative that takes into account the curvature of the spacetime
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bull From these quantities the path of any particle can be calculated This is how geometry tells
matter how to move
bull The other direction (matter tells spacetime how to curve) requires to know the distribution of
matter (massenergymomentum) described through the stress-energy tensor T
bull After many years of thinking Einstein reached a satisfactory form for the equations relating
geometry and matter
Einstein_tensor = constant x T
bull The Einstein tensor (usually called G) is a tensor in 4D spacetime that has the wanted properties
of
bull being a symmetric tensor (it must because the stress-energy tensor is symmetric)
bull having vanishing (covariant) divergence (it must because the stress-energy tensor has
vanishing divergence)
bull the weak-field limit of the Einstein equations gives the Newtonian Poisson equation (from the
comparison to which the value of the above constant is found)
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ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
The Einstein equations
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
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bull Gravitational redshift
bull change in the frequency of light moving in regions of different curvature
bull measured in experiments and taken into account by the GPS
bull Periastron shift
bull first measured in the perihelion advance of Mercury (the GR prediction coincides with
the ldquoanomalousrdquo advance if computed in Newtonian theory)bull binary pulsar (strong fields so larger shifts)
bull Bending of light
bull first measured in the bending of photons traveling near the Sun
bull gravitational lensing
bull
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bull A black hole is literally a region from which light cannot escape
bull The boundary of a black hole is called the event horizon When something passes through the
event horizon it can no longer communicate in any way with the world outside However it is
not a concrete surface there is no local measurement that can tell an observer that heshe is on
the event horizon
bull General relativity predicts that black holes can form by gravitational collapse Once a star has
burned up its supply of nuclear fuel it can no longer support itself against its own weight and it
collapses It will reach a critical density at which an explosive re-ignition of burning occurs
called a supernova
bull There are several candidate objects that are thought to be black holes (because they are very
compact) but there has been no direct observation of black holes up to now
bull The definitive way to detect black holes is through gravitational waves (more on this later)
because gravitational waves from black holes are distinct from those of other objects
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bull In general relativity disturbances in the spacetime curvature (the ldquooldrdquo ldquogravitational
fieldrdquo) propagate at the speed of light as gravitational radiation or gravitational waves
(also known as gravity waves but this term was already in use in fluid dynamics with a
different meaning so I recommend to avoid it)
bull Gravitational waves are a strong point of general relativity which solves the action-at-a-
distance problem of Newtonian gravity
bull Analogously to light which is produced by the motion of electric charges gravitational
waves are produced by the motion of anything (mass-energy) When you wave yourhand you make gravitational waves
bull The point is that the amplitude of gravitational radiation is very small
8182019 Luca Bai Otti
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T R3(GM )
part 3Qij
part t3
MR2
T 3
Mv2
T
LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
LGW = 15Gc5
part 3
Qij
part t3part 3
Qij
part t3 1048575
8182019 Luca Bai Otti
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LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
G4
c5
82times
10
minus74
cgs units
c5 27 times 10minus60 cgs units
8182019 Luca Bai Otti
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bull In fact gravitational waves have not been detected yet even if there are strong
indications that gravitational waves exist as predicted by general relativity
bull Such an indirect evidence comes from binary pulsars Two neutron stars orbiting each
other at least one of which is a pulsar The curvature is large so it is an interesting place
to study general relativistic effects (Incidentally their orbit may precess by 4 degree per
year)bull The spin axis of pulsars is not aligned with
their magnetic axis so at each rotation theyemit pulses of radio waves and we can
detect them from the Earth Pulsars behave
as very accurate clocks
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bull Actually such measurements are very precise and
confirm that the orbit is shrinking with time at a rate
in complete agreement with the emission of
gravitational waves predicted by general relativity
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radio
far-IR
mid-IR
near-IR
opticalx-ray
gamma-ray
GWs
It has happened over and over in the history of astronomy as a newldquowindowrdquo has been opened a ldquonewrdquo universe has been revealed
The same will happen with GW-astronomy
GSFCNASA
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Large-scale
Cryogenic
Gravitational-waveTelescope
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o Binary neutron stars
o Binary black-holes
o Deformed compact stars including Low Mass X-ray Binaries (LMXB)and pulsars
o Cosmological stochastic background
o Mixed binary systems
o Gravitational collapse (supernovae neutron stars)
o Extreme-Mass-Ratio Inspirals (EMRI)
o
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 27100
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TAMA-Tokyo GEO-Hannover LIGO-Livingston LIGO-Hanford VIRGO-Cascina
and LCGT will measure LL ~ h lt 10-21 with SN~1
Numericalrelativity
h
Knowledge of thewaveforms cancompensate for the
very small SN(matched-filtering) and
so enhance detection and allow for source-characterization
possible
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ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
The fundamental equations standard numerical relativity aims at solving
are the Einstein equations
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General relativity states that our World is a 4D and curvedspacetime and the Einstein equations describe its dynamics
How to solve the Einstein equations numerically
Prominently there is no a priori concept of ldquoflowing of timerdquo (we arenot involved in thermodynamics here) time is just one of thedimensions and on the same level as space dimensions
There is a successful recipe though
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We have the illusion to live in 3D and it is easier to tell computers toperform simulations (time-)step by (time-)step
Also assign a normalizationsuch that
Define therefore
So given a manifold describing a spacetime with 4-metricwe want to foliate it via space-like three-dimensionalhypersurfaces We label such hypersurfaces with the
time coordinate t
(ldquothe direction of timerdquo)
(As mostly in numer ical relativity the signature is here -+++)
The function is called the rdquolapserdquo function and itis strictly positive for spacelike hypersurfaces
α
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so that
ii) and the spatial metric
Letrsquos also define
i) the unit normal vector to the hypersurface
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The spatial part is obtained by contracting with the spatialprojection operator defined as
By using we can decompose any 4D tensor into a
purely spatial part (hence in ) and a purely timelike part (henceorthogonal to and aligned with )
while the timelike part is obtained by contracting with the timelikeprojection operator
The two projectors are obviously orthogonal
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The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariantderivative
All the 4D tensors in the Einstein equations can be projectedstraightforwardly onto the 3D spatial slice
In particular the 3D connection coefficients
the 3D Riemann tensor
and the 3D contractions of the 3D Riemann tensor ie the 3DRicci tensor the 3D Ricci scalar and
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Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic
curvatureThe extrinsic curvature is defined in terms of the unit normal to as
where is the Lie derivative along
This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric
We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic
curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold
and this can be shown to be equivalent also to
8182019 Luca Bai Otti
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Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
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Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
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W f l i EOS
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
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bull Einstein reasoned that a uniform gravitational field in some
direction is indistinguishable from a uniform acceleration in
the opposite direction
bull And so postulated his version of the equivalence principlewhich puts gravity and acceleration on an equal footing
bull ldquowe [] assume the complete physical equivalence of a
gravitational field and a corresponding acceleration of
the reference system
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bull Einstein started to think of the path of an object as a property of spacetime itself rather than
being related with the specific properties of the object
bull The idea is that gravity is a manifestation of the fact that objects in free fall follow
geodesics in curved spacetimes
bull What are geodesics
bull We know in our ordinary experience (flat spacetime) that in the absence of any forces
objects follow straight lines and we also know that straight lines are the shortest possible
paths that connect two points in such conditions
bull The generalization of the notion of a ldquostraight linerdquo valid also in curved spacetimes is called
geodesic
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bull A famous story to simply illustrate the idea of general relativity is the Parable of the Apple
by Misner Thorne and Wheeler [Gravitation (1973)]
bull The parable tries to explain the nature of gravitation in terms of the curvature of spacetime
The spacetime of the parable is the two-dimensional curved surface of an apple
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bull The tale goes like this One day a student reflecting on the difference between Einsteins and
Newtons views about gravity noticed ants running on the surface of an apple
bull By advancing alternately and of the same amount the left and right legs the ants seemed to take
the most economical path ldquowow they are going along geodesics on this surfacerdquo
bull The student followed the path of an ant tracing it and then cutting with a knife a small stripe
around the trace ldquoIndeed when put on a plane the path is a straight linerdquo
bull Each geodesic may be regarded as a path (world line) of a free particle on this surface (taken as
a two-dimensional spacetime)
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bull Then the student looked at two ants going from the same spot onto initially divergent paths but
then when approaching the top (near the dimple) of the apple the paths crossed and continued
into different directions
bull The reason of the curved trajectories is
bull According to Newton gravitation is acting at a distance from a center of attraction (the
dimple)
bull According to Einstein the local geometry of the surface around the dimple is curved
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bull Comments
bull Einstein interpretation dispenses with any action-at-a-distance
bull Although the surface of the apple is curved if you look at any local spot closely (with amagnifying glass) its geometry looks like that of a flat surface (the Minkowski spacetime of
the special relativity)
bull The interaction of spacetime and matter is summarized in John Archibald Wheelers favorite
words spacetime tells matter how to move and matter tells spacetime how to curve
bull This reciprocal influence (matter spacetime) makes Einsteinrsquos field equation non-linearand so very hard to solve
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bull Summary of the parable
bull 1) objects follow geodesics and locally geodesics appear straight
bull 2) over more extended regions of space and time geodesics originally receding from eachother begin to approach at a rate governed by the curvature of spacetime and this effect of
geometry on matter is what was called ldquogravitationrdquo
bull 3) matter in turn warps geometry
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bull Important quantities in general relativity
bull the metric (the metric tensor g) which may be regarded as a machinery for measuring
distances
dS 2 = gmicroν dxmicrodxν
Rmicroν = Rα
microαν bull the Ricci tensor and the curvature (Ricci) scalar R = gmicroν Rmicroν
Rαβmicroν = Γ
αβν micro minus Γ
αβmicroν + Γ
ασmicroΓ
σβν minus Γ
ασν Γ
σβmicro
Γαβmicro =
12
gασ(gβσmicro + gσmicroβ minus gβmicroσ)
bull curvature expressed by the Riemann curvature tensor
(where are the Christoffel symbols)
bull covariant derivative a derivative that takes into account the curvature of the spacetime
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bull From these quantities the path of any particle can be calculated This is how geometry tells
matter how to move
bull The other direction (matter tells spacetime how to curve) requires to know the distribution of
matter (massenergymomentum) described through the stress-energy tensor T
bull After many years of thinking Einstein reached a satisfactory form for the equations relating
geometry and matter
Einstein_tensor = constant x T
bull The Einstein tensor (usually called G) is a tensor in 4D spacetime that has the wanted properties
of
bull being a symmetric tensor (it must because the stress-energy tensor is symmetric)
bull having vanishing (covariant) divergence (it must because the stress-energy tensor has
vanishing divergence)
bull the weak-field limit of the Einstein equations gives the Newtonian Poisson equation (from the
comparison to which the value of the above constant is found)
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ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
The Einstein equations
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
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bull Gravitational redshift
bull change in the frequency of light moving in regions of different curvature
bull measured in experiments and taken into account by the GPS
bull Periastron shift
bull first measured in the perihelion advance of Mercury (the GR prediction coincides with
the ldquoanomalousrdquo advance if computed in Newtonian theory)bull binary pulsar (strong fields so larger shifts)
bull Bending of light
bull first measured in the bending of photons traveling near the Sun
bull gravitational lensing
bull
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bull A black hole is literally a region from which light cannot escape
bull The boundary of a black hole is called the event horizon When something passes through the
event horizon it can no longer communicate in any way with the world outside However it is
not a concrete surface there is no local measurement that can tell an observer that heshe is on
the event horizon
bull General relativity predicts that black holes can form by gravitational collapse Once a star has
burned up its supply of nuclear fuel it can no longer support itself against its own weight and it
collapses It will reach a critical density at which an explosive re-ignition of burning occurs
called a supernova
bull There are several candidate objects that are thought to be black holes (because they are very
compact) but there has been no direct observation of black holes up to now
bull The definitive way to detect black holes is through gravitational waves (more on this later)
because gravitational waves from black holes are distinct from those of other objects
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bull In general relativity disturbances in the spacetime curvature (the ldquooldrdquo ldquogravitational
fieldrdquo) propagate at the speed of light as gravitational radiation or gravitational waves
(also known as gravity waves but this term was already in use in fluid dynamics with a
different meaning so I recommend to avoid it)
bull Gravitational waves are a strong point of general relativity which solves the action-at-a-
distance problem of Newtonian gravity
bull Analogously to light which is produced by the motion of electric charges gravitational
waves are produced by the motion of anything (mass-energy) When you wave yourhand you make gravitational waves
bull The point is that the amplitude of gravitational radiation is very small
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T R3(GM )
part 3Qij
part t3
MR2
T 3
Mv2
T
LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
LGW = 15Gc5
part 3
Qij
part t3part 3
Qij
part t3 1048575
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LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
G4
c5
82times
10
minus74
cgs units
c5 27 times 10minus60 cgs units
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bull In fact gravitational waves have not been detected yet even if there are strong
indications that gravitational waves exist as predicted by general relativity
bull Such an indirect evidence comes from binary pulsars Two neutron stars orbiting each
other at least one of which is a pulsar The curvature is large so it is an interesting place
to study general relativistic effects (Incidentally their orbit may precess by 4 degree per
year)bull The spin axis of pulsars is not aligned with
their magnetic axis so at each rotation theyemit pulses of radio waves and we can
detect them from the Earth Pulsars behave
as very accurate clocks
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bull Actually such measurements are very precise and
confirm that the orbit is shrinking with time at a rate
in complete agreement with the emission of
gravitational waves predicted by general relativity
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radio
far-IR
mid-IR
near-IR
opticalx-ray
gamma-ray
GWs
It has happened over and over in the history of astronomy as a newldquowindowrdquo has been opened a ldquonewrdquo universe has been revealed
The same will happen with GW-astronomy
GSFCNASA
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Large-scale
Cryogenic
Gravitational-waveTelescope
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o Binary neutron stars
o Binary black-holes
o Deformed compact stars including Low Mass X-ray Binaries (LMXB)and pulsars
o Cosmological stochastic background
o Mixed binary systems
o Gravitational collapse (supernovae neutron stars)
o Extreme-Mass-Ratio Inspirals (EMRI)
o
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TAMA-Tokyo GEO-Hannover LIGO-Livingston LIGO-Hanford VIRGO-Cascina
and LCGT will measure LL ~ h lt 10-21 with SN~1
Numericalrelativity
h
Knowledge of thewaveforms cancompensate for the
very small SN(matched-filtering) and
so enhance detection and allow for source-characterization
possible
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ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
The fundamental equations standard numerical relativity aims at solving
are the Einstein equations
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General relativity states that our World is a 4D and curvedspacetime and the Einstein equations describe its dynamics
How to solve the Einstein equations numerically
Prominently there is no a priori concept of ldquoflowing of timerdquo (we arenot involved in thermodynamics here) time is just one of thedimensions and on the same level as space dimensions
There is a successful recipe though
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We have the illusion to live in 3D and it is easier to tell computers toperform simulations (time-)step by (time-)step
Also assign a normalizationsuch that
Define therefore
So given a manifold describing a spacetime with 4-metricwe want to foliate it via space-like three-dimensionalhypersurfaces We label such hypersurfaces with the
time coordinate t
(ldquothe direction of timerdquo)
(As mostly in numer ical relativity the signature is here -+++)
The function is called the rdquolapserdquo function and itis strictly positive for spacelike hypersurfaces
α
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so that
ii) and the spatial metric
Letrsquos also define
i) the unit normal vector to the hypersurface
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The spatial part is obtained by contracting with the spatialprojection operator defined as
By using we can decompose any 4D tensor into a
purely spatial part (hence in ) and a purely timelike part (henceorthogonal to and aligned with )
while the timelike part is obtained by contracting with the timelikeprojection operator
The two projectors are obviously orthogonal
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The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariantderivative
All the 4D tensors in the Einstein equations can be projectedstraightforwardly onto the 3D spatial slice
In particular the 3D connection coefficients
the 3D Riemann tensor
and the 3D contractions of the 3D Riemann tensor ie the 3DRicci tensor the 3D Ricci scalar and
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Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic
curvatureThe extrinsic curvature is defined in terms of the unit normal to as
where is the Lie derivative along
This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric
We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic
curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold
and this can be shown to be equivalent also to
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Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 61100
There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
8182019 Luca Bai Otti
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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httpslidepdfcomreaderfullluca-bai-otti 64100
We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 65100
Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 66100
where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 67100
conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 68100
Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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httpslidepdfcomreaderfullluca-bai-otti 71100
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 73100
In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
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W f l i EOS
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
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bull Einstein started to think of the path of an object as a property of spacetime itself rather than
being related with the specific properties of the object
bull The idea is that gravity is a manifestation of the fact that objects in free fall follow
geodesics in curved spacetimes
bull What are geodesics
bull We know in our ordinary experience (flat spacetime) that in the absence of any forces
objects follow straight lines and we also know that straight lines are the shortest possible
paths that connect two points in such conditions
bull The generalization of the notion of a ldquostraight linerdquo valid also in curved spacetimes is called
geodesic
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bull A famous story to simply illustrate the idea of general relativity is the Parable of the Apple
by Misner Thorne and Wheeler [Gravitation (1973)]
bull The parable tries to explain the nature of gravitation in terms of the curvature of spacetime
The spacetime of the parable is the two-dimensional curved surface of an apple
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bull The tale goes like this One day a student reflecting on the difference between Einsteins and
Newtons views about gravity noticed ants running on the surface of an apple
bull By advancing alternately and of the same amount the left and right legs the ants seemed to take
the most economical path ldquowow they are going along geodesics on this surfacerdquo
bull The student followed the path of an ant tracing it and then cutting with a knife a small stripe
around the trace ldquoIndeed when put on a plane the path is a straight linerdquo
bull Each geodesic may be regarded as a path (world line) of a free particle on this surface (taken as
a two-dimensional spacetime)
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bull Then the student looked at two ants going from the same spot onto initially divergent paths but
then when approaching the top (near the dimple) of the apple the paths crossed and continued
into different directions
bull The reason of the curved trajectories is
bull According to Newton gravitation is acting at a distance from a center of attraction (the
dimple)
bull According to Einstein the local geometry of the surface around the dimple is curved
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bull Comments
bull Einstein interpretation dispenses with any action-at-a-distance
bull Although the surface of the apple is curved if you look at any local spot closely (with amagnifying glass) its geometry looks like that of a flat surface (the Minkowski spacetime of
the special relativity)
bull The interaction of spacetime and matter is summarized in John Archibald Wheelers favorite
words spacetime tells matter how to move and matter tells spacetime how to curve
bull This reciprocal influence (matter spacetime) makes Einsteinrsquos field equation non-linearand so very hard to solve
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bull Summary of the parable
bull 1) objects follow geodesics and locally geodesics appear straight
bull 2) over more extended regions of space and time geodesics originally receding from eachother begin to approach at a rate governed by the curvature of spacetime and this effect of
geometry on matter is what was called ldquogravitationrdquo
bull 3) matter in turn warps geometry
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bull Important quantities in general relativity
bull the metric (the metric tensor g) which may be regarded as a machinery for measuring
distances
dS 2 = gmicroν dxmicrodxν
Rmicroν = Rα
microαν bull the Ricci tensor and the curvature (Ricci) scalar R = gmicroν Rmicroν
Rαβmicroν = Γ
αβν micro minus Γ
αβmicroν + Γ
ασmicroΓ
σβν minus Γ
ασν Γ
σβmicro
Γαβmicro =
12
gασ(gβσmicro + gσmicroβ minus gβmicroσ)
bull curvature expressed by the Riemann curvature tensor
(where are the Christoffel symbols)
bull covariant derivative a derivative that takes into account the curvature of the spacetime
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bull From these quantities the path of any particle can be calculated This is how geometry tells
matter how to move
bull The other direction (matter tells spacetime how to curve) requires to know the distribution of
matter (massenergymomentum) described through the stress-energy tensor T
bull After many years of thinking Einstein reached a satisfactory form for the equations relating
geometry and matter
Einstein_tensor = constant x T
bull The Einstein tensor (usually called G) is a tensor in 4D spacetime that has the wanted properties
of
bull being a symmetric tensor (it must because the stress-energy tensor is symmetric)
bull having vanishing (covariant) divergence (it must because the stress-energy tensor has
vanishing divergence)
bull the weak-field limit of the Einstein equations gives the Newtonian Poisson equation (from the
comparison to which the value of the above constant is found)
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ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
The Einstein equations
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
8182019 Luca Bai Otti
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bull Gravitational redshift
bull change in the frequency of light moving in regions of different curvature
bull measured in experiments and taken into account by the GPS
bull Periastron shift
bull first measured in the perihelion advance of Mercury (the GR prediction coincides with
the ldquoanomalousrdquo advance if computed in Newtonian theory)bull binary pulsar (strong fields so larger shifts)
bull Bending of light
bull first measured in the bending of photons traveling near the Sun
bull gravitational lensing
bull
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bull A black hole is literally a region from which light cannot escape
bull The boundary of a black hole is called the event horizon When something passes through the
event horizon it can no longer communicate in any way with the world outside However it is
not a concrete surface there is no local measurement that can tell an observer that heshe is on
the event horizon
bull General relativity predicts that black holes can form by gravitational collapse Once a star has
burned up its supply of nuclear fuel it can no longer support itself against its own weight and it
collapses It will reach a critical density at which an explosive re-ignition of burning occurs
called a supernova
bull There are several candidate objects that are thought to be black holes (because they are very
compact) but there has been no direct observation of black holes up to now
bull The definitive way to detect black holes is through gravitational waves (more on this later)
because gravitational waves from black holes are distinct from those of other objects
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bull In general relativity disturbances in the spacetime curvature (the ldquooldrdquo ldquogravitational
fieldrdquo) propagate at the speed of light as gravitational radiation or gravitational waves
(also known as gravity waves but this term was already in use in fluid dynamics with a
different meaning so I recommend to avoid it)
bull Gravitational waves are a strong point of general relativity which solves the action-at-a-
distance problem of Newtonian gravity
bull Analogously to light which is produced by the motion of electric charges gravitational
waves are produced by the motion of anything (mass-energy) When you wave yourhand you make gravitational waves
bull The point is that the amplitude of gravitational radiation is very small
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T R3(GM )
part 3Qij
part t3
MR2
T 3
Mv2
T
LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
LGW = 15Gc5
part 3
Qij
part t3part 3
Qij
part t3 1048575
8182019 Luca Bai Otti
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LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
G4
c5
82times
10
minus74
cgs units
c5 27 times 10minus60 cgs units
8182019 Luca Bai Otti
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bull In fact gravitational waves have not been detected yet even if there are strong
indications that gravitational waves exist as predicted by general relativity
bull Such an indirect evidence comes from binary pulsars Two neutron stars orbiting each
other at least one of which is a pulsar The curvature is large so it is an interesting place
to study general relativistic effects (Incidentally their orbit may precess by 4 degree per
year)bull The spin axis of pulsars is not aligned with
their magnetic axis so at each rotation theyemit pulses of radio waves and we can
detect them from the Earth Pulsars behave
as very accurate clocks
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bull Actually such measurements are very precise and
confirm that the orbit is shrinking with time at a rate
in complete agreement with the emission of
gravitational waves predicted by general relativity
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radio
far-IR
mid-IR
near-IR
opticalx-ray
gamma-ray
GWs
It has happened over and over in the history of astronomy as a newldquowindowrdquo has been opened a ldquonewrdquo universe has been revealed
The same will happen with GW-astronomy
GSFCNASA
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Large-scale
Cryogenic
Gravitational-waveTelescope
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o Binary neutron stars
o Binary black-holes
o Deformed compact stars including Low Mass X-ray Binaries (LMXB)and pulsars
o Cosmological stochastic background
o Mixed binary systems
o Gravitational collapse (supernovae neutron stars)
o Extreme-Mass-Ratio Inspirals (EMRI)
o
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TAMA-Tokyo GEO-Hannover LIGO-Livingston LIGO-Hanford VIRGO-Cascina
and LCGT will measure LL ~ h lt 10-21 with SN~1
Numericalrelativity
h
Knowledge of thewaveforms cancompensate for the
very small SN(matched-filtering) and
so enhance detection and allow for source-characterization
possible
8182019 Luca Bai Otti
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ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
The fundamental equations standard numerical relativity aims at solving
are the Einstein equations
8182019 Luca Bai Otti
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General relativity states that our World is a 4D and curvedspacetime and the Einstein equations describe its dynamics
How to solve the Einstein equations numerically
Prominently there is no a priori concept of ldquoflowing of timerdquo (we arenot involved in thermodynamics here) time is just one of thedimensions and on the same level as space dimensions
There is a successful recipe though
8182019 Luca Bai Otti
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We have the illusion to live in 3D and it is easier to tell computers toperform simulations (time-)step by (time-)step
Also assign a normalizationsuch that
Define therefore
So given a manifold describing a spacetime with 4-metricwe want to foliate it via space-like three-dimensionalhypersurfaces We label such hypersurfaces with the
time coordinate t
(ldquothe direction of timerdquo)
(As mostly in numer ical relativity the signature is here -+++)
The function is called the rdquolapserdquo function and itis strictly positive for spacelike hypersurfaces
α
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so that
ii) and the spatial metric
Letrsquos also define
i) the unit normal vector to the hypersurface
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The spatial part is obtained by contracting with the spatialprojection operator defined as
By using we can decompose any 4D tensor into a
purely spatial part (hence in ) and a purely timelike part (henceorthogonal to and aligned with )
while the timelike part is obtained by contracting with the timelikeprojection operator
The two projectors are obviously orthogonal
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The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariantderivative
All the 4D tensors in the Einstein equations can be projectedstraightforwardly onto the 3D spatial slice
In particular the 3D connection coefficients
the 3D Riemann tensor
and the 3D contractions of the 3D Riemann tensor ie the 3DRicci tensor the 3D Ricci scalar and
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Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic
curvatureThe extrinsic curvature is defined in terms of the unit normal to as
where is the Lie derivative along
This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric
We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic
curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold
and this can be shown to be equivalent also to
8182019 Luca Bai Otti
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Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
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httpslidepdfcomreaderfullluca-bai-otti 37100
Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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httpslidepdfcomreaderfullluca-bai-otti 38100
Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
8182019 Luca Bai Otti
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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httpslidepdfcomreaderfullluca-bai-otti 46100
We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
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Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
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W f l i EOS
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
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bull A famous story to simply illustrate the idea of general relativity is the Parable of the Apple
by Misner Thorne and Wheeler [Gravitation (1973)]
bull The parable tries to explain the nature of gravitation in terms of the curvature of spacetime
The spacetime of the parable is the two-dimensional curved surface of an apple
8182019 Luca Bai Otti
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bull The tale goes like this One day a student reflecting on the difference between Einsteins and
Newtons views about gravity noticed ants running on the surface of an apple
bull By advancing alternately and of the same amount the left and right legs the ants seemed to take
the most economical path ldquowow they are going along geodesics on this surfacerdquo
bull The student followed the path of an ant tracing it and then cutting with a knife a small stripe
around the trace ldquoIndeed when put on a plane the path is a straight linerdquo
bull Each geodesic may be regarded as a path (world line) of a free particle on this surface (taken as
a two-dimensional spacetime)
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bull Then the student looked at two ants going from the same spot onto initially divergent paths but
then when approaching the top (near the dimple) of the apple the paths crossed and continued
into different directions
bull The reason of the curved trajectories is
bull According to Newton gravitation is acting at a distance from a center of attraction (the
dimple)
bull According to Einstein the local geometry of the surface around the dimple is curved
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bull Comments
bull Einstein interpretation dispenses with any action-at-a-distance
bull Although the surface of the apple is curved if you look at any local spot closely (with amagnifying glass) its geometry looks like that of a flat surface (the Minkowski spacetime of
the special relativity)
bull The interaction of spacetime and matter is summarized in John Archibald Wheelers favorite
words spacetime tells matter how to move and matter tells spacetime how to curve
bull This reciprocal influence (matter spacetime) makes Einsteinrsquos field equation non-linearand so very hard to solve
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bull Summary of the parable
bull 1) objects follow geodesics and locally geodesics appear straight
bull 2) over more extended regions of space and time geodesics originally receding from eachother begin to approach at a rate governed by the curvature of spacetime and this effect of
geometry on matter is what was called ldquogravitationrdquo
bull 3) matter in turn warps geometry
8182019 Luca Bai Otti
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bull Important quantities in general relativity
bull the metric (the metric tensor g) which may be regarded as a machinery for measuring
distances
dS 2 = gmicroν dxmicrodxν
Rmicroν = Rα
microαν bull the Ricci tensor and the curvature (Ricci) scalar R = gmicroν Rmicroν
Rαβmicroν = Γ
αβν micro minus Γ
αβmicroν + Γ
ασmicroΓ
σβν minus Γ
ασν Γ
σβmicro
Γαβmicro =
12
gασ(gβσmicro + gσmicroβ minus gβmicroσ)
bull curvature expressed by the Riemann curvature tensor
(where are the Christoffel symbols)
bull covariant derivative a derivative that takes into account the curvature of the spacetime
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bull From these quantities the path of any particle can be calculated This is how geometry tells
matter how to move
bull The other direction (matter tells spacetime how to curve) requires to know the distribution of
matter (massenergymomentum) described through the stress-energy tensor T
bull After many years of thinking Einstein reached a satisfactory form for the equations relating
geometry and matter
Einstein_tensor = constant x T
bull The Einstein tensor (usually called G) is a tensor in 4D spacetime that has the wanted properties
of
bull being a symmetric tensor (it must because the stress-energy tensor is symmetric)
bull having vanishing (covariant) divergence (it must because the stress-energy tensor has
vanishing divergence)
bull the weak-field limit of the Einstein equations gives the Newtonian Poisson equation (from the
comparison to which the value of the above constant is found)
8182019 Luca Bai Otti
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ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
The Einstein equations
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
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bull Gravitational redshift
bull change in the frequency of light moving in regions of different curvature
bull measured in experiments and taken into account by the GPS
bull Periastron shift
bull first measured in the perihelion advance of Mercury (the GR prediction coincides with
the ldquoanomalousrdquo advance if computed in Newtonian theory)bull binary pulsar (strong fields so larger shifts)
bull Bending of light
bull first measured in the bending of photons traveling near the Sun
bull gravitational lensing
bull
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bull A black hole is literally a region from which light cannot escape
bull The boundary of a black hole is called the event horizon When something passes through the
event horizon it can no longer communicate in any way with the world outside However it is
not a concrete surface there is no local measurement that can tell an observer that heshe is on
the event horizon
bull General relativity predicts that black holes can form by gravitational collapse Once a star has
burned up its supply of nuclear fuel it can no longer support itself against its own weight and it
collapses It will reach a critical density at which an explosive re-ignition of burning occurs
called a supernova
bull There are several candidate objects that are thought to be black holes (because they are very
compact) but there has been no direct observation of black holes up to now
bull The definitive way to detect black holes is through gravitational waves (more on this later)
because gravitational waves from black holes are distinct from those of other objects
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bull In general relativity disturbances in the spacetime curvature (the ldquooldrdquo ldquogravitational
fieldrdquo) propagate at the speed of light as gravitational radiation or gravitational waves
(also known as gravity waves but this term was already in use in fluid dynamics with a
different meaning so I recommend to avoid it)
bull Gravitational waves are a strong point of general relativity which solves the action-at-a-
distance problem of Newtonian gravity
bull Analogously to light which is produced by the motion of electric charges gravitational
waves are produced by the motion of anything (mass-energy) When you wave yourhand you make gravitational waves
bull The point is that the amplitude of gravitational radiation is very small
8182019 Luca Bai Otti
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T R3(GM )
part 3Qij
part t3
MR2
T 3
Mv2
T
LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
LGW = 15Gc5
part 3
Qij
part t3part 3
Qij
part t3 1048575
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 19100
LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
G4
c5
82times
10
minus74
cgs units
c5 27 times 10minus60 cgs units
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 20100
bull In fact gravitational waves have not been detected yet even if there are strong
indications that gravitational waves exist as predicted by general relativity
bull Such an indirect evidence comes from binary pulsars Two neutron stars orbiting each
other at least one of which is a pulsar The curvature is large so it is an interesting place
to study general relativistic effects (Incidentally their orbit may precess by 4 degree per
year)bull The spin axis of pulsars is not aligned with
their magnetic axis so at each rotation theyemit pulses of radio waves and we can
detect them from the Earth Pulsars behave
as very accurate clocks
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bull Actually such measurements are very precise and
confirm that the orbit is shrinking with time at a rate
in complete agreement with the emission of
gravitational waves predicted by general relativity
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radio
far-IR
mid-IR
near-IR
opticalx-ray
gamma-ray
GWs
It has happened over and over in the history of astronomy as a newldquowindowrdquo has been opened a ldquonewrdquo universe has been revealed
The same will happen with GW-astronomy
GSFCNASA
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Large-scale
Cryogenic
Gravitational-waveTelescope
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o Binary neutron stars
o Binary black-holes
o Deformed compact stars including Low Mass X-ray Binaries (LMXB)and pulsars
o Cosmological stochastic background
o Mixed binary systems
o Gravitational collapse (supernovae neutron stars)
o Extreme-Mass-Ratio Inspirals (EMRI)
o
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TAMA-Tokyo GEO-Hannover LIGO-Livingston LIGO-Hanford VIRGO-Cascina
and LCGT will measure LL ~ h lt 10-21 with SN~1
Numericalrelativity
h
Knowledge of thewaveforms cancompensate for the
very small SN(matched-filtering) and
so enhance detection and allow for source-characterization
possible
8182019 Luca Bai Otti
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ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
The fundamental equations standard numerical relativity aims at solving
are the Einstein equations
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General relativity states that our World is a 4D and curvedspacetime and the Einstein equations describe its dynamics
How to solve the Einstein equations numerically
Prominently there is no a priori concept of ldquoflowing of timerdquo (we arenot involved in thermodynamics here) time is just one of thedimensions and on the same level as space dimensions
There is a successful recipe though
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We have the illusion to live in 3D and it is easier to tell computers toperform simulations (time-)step by (time-)step
Also assign a normalizationsuch that
Define therefore
So given a manifold describing a spacetime with 4-metricwe want to foliate it via space-like three-dimensionalhypersurfaces We label such hypersurfaces with the
time coordinate t
(ldquothe direction of timerdquo)
(As mostly in numer ical relativity the signature is here -+++)
The function is called the rdquolapserdquo function and itis strictly positive for spacelike hypersurfaces
α
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so that
ii) and the spatial metric
Letrsquos also define
i) the unit normal vector to the hypersurface
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The spatial part is obtained by contracting with the spatialprojection operator defined as
By using we can decompose any 4D tensor into a
purely spatial part (hence in ) and a purely timelike part (henceorthogonal to and aligned with )
while the timelike part is obtained by contracting with the timelikeprojection operator
The two projectors are obviously orthogonal
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The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariantderivative
All the 4D tensors in the Einstein equations can be projectedstraightforwardly onto the 3D spatial slice
In particular the 3D connection coefficients
the 3D Riemann tensor
and the 3D contractions of the 3D Riemann tensor ie the 3DRicci tensor the 3D Ricci scalar and
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Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic
curvatureThe extrinsic curvature is defined in terms of the unit normal to as
where is the Lie derivative along
This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric
We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic
curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold
and this can be shown to be equivalent also to
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Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
8182019 Luca Bai Otti
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
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Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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httpslidepdfcomreaderfullluca-bai-otti 71100
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
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W f l i EOS
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
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bull The tale goes like this One day a student reflecting on the difference between Einsteins and
Newtons views about gravity noticed ants running on the surface of an apple
bull By advancing alternately and of the same amount the left and right legs the ants seemed to take
the most economical path ldquowow they are going along geodesics on this surfacerdquo
bull The student followed the path of an ant tracing it and then cutting with a knife a small stripe
around the trace ldquoIndeed when put on a plane the path is a straight linerdquo
bull Each geodesic may be regarded as a path (world line) of a free particle on this surface (taken as
a two-dimensional spacetime)
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bull Then the student looked at two ants going from the same spot onto initially divergent paths but
then when approaching the top (near the dimple) of the apple the paths crossed and continued
into different directions
bull The reason of the curved trajectories is
bull According to Newton gravitation is acting at a distance from a center of attraction (the
dimple)
bull According to Einstein the local geometry of the surface around the dimple is curved
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bull Comments
bull Einstein interpretation dispenses with any action-at-a-distance
bull Although the surface of the apple is curved if you look at any local spot closely (with amagnifying glass) its geometry looks like that of a flat surface (the Minkowski spacetime of
the special relativity)
bull The interaction of spacetime and matter is summarized in John Archibald Wheelers favorite
words spacetime tells matter how to move and matter tells spacetime how to curve
bull This reciprocal influence (matter spacetime) makes Einsteinrsquos field equation non-linearand so very hard to solve
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bull Summary of the parable
bull 1) objects follow geodesics and locally geodesics appear straight
bull 2) over more extended regions of space and time geodesics originally receding from eachother begin to approach at a rate governed by the curvature of spacetime and this effect of
geometry on matter is what was called ldquogravitationrdquo
bull 3) matter in turn warps geometry
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bull Important quantities in general relativity
bull the metric (the metric tensor g) which may be regarded as a machinery for measuring
distances
dS 2 = gmicroν dxmicrodxν
Rmicroν = Rα
microαν bull the Ricci tensor and the curvature (Ricci) scalar R = gmicroν Rmicroν
Rαβmicroν = Γ
αβν micro minus Γ
αβmicroν + Γ
ασmicroΓ
σβν minus Γ
ασν Γ
σβmicro
Γαβmicro =
12
gασ(gβσmicro + gσmicroβ minus gβmicroσ)
bull curvature expressed by the Riemann curvature tensor
(where are the Christoffel symbols)
bull covariant derivative a derivative that takes into account the curvature of the spacetime
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bull From these quantities the path of any particle can be calculated This is how geometry tells
matter how to move
bull The other direction (matter tells spacetime how to curve) requires to know the distribution of
matter (massenergymomentum) described through the stress-energy tensor T
bull After many years of thinking Einstein reached a satisfactory form for the equations relating
geometry and matter
Einstein_tensor = constant x T
bull The Einstein tensor (usually called G) is a tensor in 4D spacetime that has the wanted properties
of
bull being a symmetric tensor (it must because the stress-energy tensor is symmetric)
bull having vanishing (covariant) divergence (it must because the stress-energy tensor has
vanishing divergence)
bull the weak-field limit of the Einstein equations gives the Newtonian Poisson equation (from the
comparison to which the value of the above constant is found)
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ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
The Einstein equations
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
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bull Gravitational redshift
bull change in the frequency of light moving in regions of different curvature
bull measured in experiments and taken into account by the GPS
bull Periastron shift
bull first measured in the perihelion advance of Mercury (the GR prediction coincides with
the ldquoanomalousrdquo advance if computed in Newtonian theory)bull binary pulsar (strong fields so larger shifts)
bull Bending of light
bull first measured in the bending of photons traveling near the Sun
bull gravitational lensing
bull
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bull A black hole is literally a region from which light cannot escape
bull The boundary of a black hole is called the event horizon When something passes through the
event horizon it can no longer communicate in any way with the world outside However it is
not a concrete surface there is no local measurement that can tell an observer that heshe is on
the event horizon
bull General relativity predicts that black holes can form by gravitational collapse Once a star has
burned up its supply of nuclear fuel it can no longer support itself against its own weight and it
collapses It will reach a critical density at which an explosive re-ignition of burning occurs
called a supernova
bull There are several candidate objects that are thought to be black holes (because they are very
compact) but there has been no direct observation of black holes up to now
bull The definitive way to detect black holes is through gravitational waves (more on this later)
because gravitational waves from black holes are distinct from those of other objects
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bull In general relativity disturbances in the spacetime curvature (the ldquooldrdquo ldquogravitational
fieldrdquo) propagate at the speed of light as gravitational radiation or gravitational waves
(also known as gravity waves but this term was already in use in fluid dynamics with a
different meaning so I recommend to avoid it)
bull Gravitational waves are a strong point of general relativity which solves the action-at-a-
distance problem of Newtonian gravity
bull Analogously to light which is produced by the motion of electric charges gravitational
waves are produced by the motion of anything (mass-energy) When you wave yourhand you make gravitational waves
bull The point is that the amplitude of gravitational radiation is very small
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T R3(GM )
part 3Qij
part t3
MR2
T 3
Mv2
T
LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
LGW = 15Gc5
part 3
Qij
part t3part 3
Qij
part t3 1048575
8182019 Luca Bai Otti
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LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
G4
c5
82times
10
minus74
cgs units
c5 27 times 10minus60 cgs units
8182019 Luca Bai Otti
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bull In fact gravitational waves have not been detected yet even if there are strong
indications that gravitational waves exist as predicted by general relativity
bull Such an indirect evidence comes from binary pulsars Two neutron stars orbiting each
other at least one of which is a pulsar The curvature is large so it is an interesting place
to study general relativistic effects (Incidentally their orbit may precess by 4 degree per
year)bull The spin axis of pulsars is not aligned with
their magnetic axis so at each rotation theyemit pulses of radio waves and we can
detect them from the Earth Pulsars behave
as very accurate clocks
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bull Actually such measurements are very precise and
confirm that the orbit is shrinking with time at a rate
in complete agreement with the emission of
gravitational waves predicted by general relativity
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radio
far-IR
mid-IR
near-IR
opticalx-ray
gamma-ray
GWs
It has happened over and over in the history of astronomy as a newldquowindowrdquo has been opened a ldquonewrdquo universe has been revealed
The same will happen with GW-astronomy
GSFCNASA
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Large-scale
Cryogenic
Gravitational-waveTelescope
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o Binary neutron stars
o Binary black-holes
o Deformed compact stars including Low Mass X-ray Binaries (LMXB)and pulsars
o Cosmological stochastic background
o Mixed binary systems
o Gravitational collapse (supernovae neutron stars)
o Extreme-Mass-Ratio Inspirals (EMRI)
o
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TAMA-Tokyo GEO-Hannover LIGO-Livingston LIGO-Hanford VIRGO-Cascina
and LCGT will measure LL ~ h lt 10-21 with SN~1
Numericalrelativity
h
Knowledge of thewaveforms cancompensate for the
very small SN(matched-filtering) and
so enhance detection and allow for source-characterization
possible
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ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
The fundamental equations standard numerical relativity aims at solving
are the Einstein equations
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General relativity states that our World is a 4D and curvedspacetime and the Einstein equations describe its dynamics
How to solve the Einstein equations numerically
Prominently there is no a priori concept of ldquoflowing of timerdquo (we arenot involved in thermodynamics here) time is just one of thedimensions and on the same level as space dimensions
There is a successful recipe though
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We have the illusion to live in 3D and it is easier to tell computers toperform simulations (time-)step by (time-)step
Also assign a normalizationsuch that
Define therefore
So given a manifold describing a spacetime with 4-metricwe want to foliate it via space-like three-dimensionalhypersurfaces We label such hypersurfaces with the
time coordinate t
(ldquothe direction of timerdquo)
(As mostly in numer ical relativity the signature is here -+++)
The function is called the rdquolapserdquo function and itis strictly positive for spacelike hypersurfaces
α
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so that
ii) and the spatial metric
Letrsquos also define
i) the unit normal vector to the hypersurface
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The spatial part is obtained by contracting with the spatialprojection operator defined as
By using we can decompose any 4D tensor into a
purely spatial part (hence in ) and a purely timelike part (henceorthogonal to and aligned with )
while the timelike part is obtained by contracting with the timelikeprojection operator
The two projectors are obviously orthogonal
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The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariantderivative
All the 4D tensors in the Einstein equations can be projectedstraightforwardly onto the 3D spatial slice
In particular the 3D connection coefficients
the 3D Riemann tensor
and the 3D contractions of the 3D Riemann tensor ie the 3DRicci tensor the 3D Ricci scalar and
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Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic
curvatureThe extrinsic curvature is defined in terms of the unit normal to as
where is the Lie derivative along
This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric
We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic
curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold
and this can be shown to be equivalent also to
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Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
8182019 Luca Bai Otti
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
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Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
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W f l i EOS
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
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bull Then the student looked at two ants going from the same spot onto initially divergent paths but
then when approaching the top (near the dimple) of the apple the paths crossed and continued
into different directions
bull The reason of the curved trajectories is
bull According to Newton gravitation is acting at a distance from a center of attraction (the
dimple)
bull According to Einstein the local geometry of the surface around the dimple is curved
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bull Comments
bull Einstein interpretation dispenses with any action-at-a-distance
bull Although the surface of the apple is curved if you look at any local spot closely (with amagnifying glass) its geometry looks like that of a flat surface (the Minkowski spacetime of
the special relativity)
bull The interaction of spacetime and matter is summarized in John Archibald Wheelers favorite
words spacetime tells matter how to move and matter tells spacetime how to curve
bull This reciprocal influence (matter spacetime) makes Einsteinrsquos field equation non-linearand so very hard to solve
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bull Summary of the parable
bull 1) objects follow geodesics and locally geodesics appear straight
bull 2) over more extended regions of space and time geodesics originally receding from eachother begin to approach at a rate governed by the curvature of spacetime and this effect of
geometry on matter is what was called ldquogravitationrdquo
bull 3) matter in turn warps geometry
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bull Important quantities in general relativity
bull the metric (the metric tensor g) which may be regarded as a machinery for measuring
distances
dS 2 = gmicroν dxmicrodxν
Rmicroν = Rα
microαν bull the Ricci tensor and the curvature (Ricci) scalar R = gmicroν Rmicroν
Rαβmicroν = Γ
αβν micro minus Γ
αβmicroν + Γ
ασmicroΓ
σβν minus Γ
ασν Γ
σβmicro
Γαβmicro =
12
gασ(gβσmicro + gσmicroβ minus gβmicroσ)
bull curvature expressed by the Riemann curvature tensor
(where are the Christoffel symbols)
bull covariant derivative a derivative that takes into account the curvature of the spacetime
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 13100
bull From these quantities the path of any particle can be calculated This is how geometry tells
matter how to move
bull The other direction (matter tells spacetime how to curve) requires to know the distribution of
matter (massenergymomentum) described through the stress-energy tensor T
bull After many years of thinking Einstein reached a satisfactory form for the equations relating
geometry and matter
Einstein_tensor = constant x T
bull The Einstein tensor (usually called G) is a tensor in 4D spacetime that has the wanted properties
of
bull being a symmetric tensor (it must because the stress-energy tensor is symmetric)
bull having vanishing (covariant) divergence (it must because the stress-energy tensor has
vanishing divergence)
bull the weak-field limit of the Einstein equations gives the Newtonian Poisson equation (from the
comparison to which the value of the above constant is found)
8182019 Luca Bai Otti
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ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
The Einstein equations
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 15100
bull Gravitational redshift
bull change in the frequency of light moving in regions of different curvature
bull measured in experiments and taken into account by the GPS
bull Periastron shift
bull first measured in the perihelion advance of Mercury (the GR prediction coincides with
the ldquoanomalousrdquo advance if computed in Newtonian theory)bull binary pulsar (strong fields so larger shifts)
bull Bending of light
bull first measured in the bending of photons traveling near the Sun
bull gravitational lensing
bull
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 16100
bull A black hole is literally a region from which light cannot escape
bull The boundary of a black hole is called the event horizon When something passes through the
event horizon it can no longer communicate in any way with the world outside However it is
not a concrete surface there is no local measurement that can tell an observer that heshe is on
the event horizon
bull General relativity predicts that black holes can form by gravitational collapse Once a star has
burned up its supply of nuclear fuel it can no longer support itself against its own weight and it
collapses It will reach a critical density at which an explosive re-ignition of burning occurs
called a supernova
bull There are several candidate objects that are thought to be black holes (because they are very
compact) but there has been no direct observation of black holes up to now
bull The definitive way to detect black holes is through gravitational waves (more on this later)
because gravitational waves from black holes are distinct from those of other objects
8182019 Luca Bai Otti
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bull In general relativity disturbances in the spacetime curvature (the ldquooldrdquo ldquogravitational
fieldrdquo) propagate at the speed of light as gravitational radiation or gravitational waves
(also known as gravity waves but this term was already in use in fluid dynamics with a
different meaning so I recommend to avoid it)
bull Gravitational waves are a strong point of general relativity which solves the action-at-a-
distance problem of Newtonian gravity
bull Analogously to light which is produced by the motion of electric charges gravitational
waves are produced by the motion of anything (mass-energy) When you wave yourhand you make gravitational waves
bull The point is that the amplitude of gravitational radiation is very small
8182019 Luca Bai Otti
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T R3(GM )
part 3Qij
part t3
MR2
T 3
Mv2
T
LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
LGW = 15Gc5
part 3
Qij
part t3part 3
Qij
part t3 1048575
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 19100
LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
G4
c5
82times
10
minus74
cgs units
c5 27 times 10minus60 cgs units
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 20100
bull In fact gravitational waves have not been detected yet even if there are strong
indications that gravitational waves exist as predicted by general relativity
bull Such an indirect evidence comes from binary pulsars Two neutron stars orbiting each
other at least one of which is a pulsar The curvature is large so it is an interesting place
to study general relativistic effects (Incidentally their orbit may precess by 4 degree per
year)bull The spin axis of pulsars is not aligned with
their magnetic axis so at each rotation theyemit pulses of radio waves and we can
detect them from the Earth Pulsars behave
as very accurate clocks
8182019 Luca Bai Otti
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bull Actually such measurements are very precise and
confirm that the orbit is shrinking with time at a rate
in complete agreement with the emission of
gravitational waves predicted by general relativity
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httpslidepdfcomreaderfullluca-bai-otti 22100
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radio
far-IR
mid-IR
near-IR
opticalx-ray
gamma-ray
GWs
It has happened over and over in the history of astronomy as a newldquowindowrdquo has been opened a ldquonewrdquo universe has been revealed
The same will happen with GW-astronomy
GSFCNASA
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 24100
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httpslidepdfcomreaderfullluca-bai-otti 25100
Large-scale
Cryogenic
Gravitational-waveTelescope
8182019 Luca Bai Otti
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o Binary neutron stars
o Binary black-holes
o Deformed compact stars including Low Mass X-ray Binaries (LMXB)and pulsars
o Cosmological stochastic background
o Mixed binary systems
o Gravitational collapse (supernovae neutron stars)
o Extreme-Mass-Ratio Inspirals (EMRI)
o
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TAMA-Tokyo GEO-Hannover LIGO-Livingston LIGO-Hanford VIRGO-Cascina
and LCGT will measure LL ~ h lt 10-21 with SN~1
Numericalrelativity
h
Knowledge of thewaveforms cancompensate for the
very small SN(matched-filtering) and
so enhance detection and allow for source-characterization
possible
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 29100
ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
The fundamental equations standard numerical relativity aims at solving
are the Einstein equations
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httpslidepdfcomreaderfullluca-bai-otti 30100
General relativity states that our World is a 4D and curvedspacetime and the Einstein equations describe its dynamics
How to solve the Einstein equations numerically
Prominently there is no a priori concept of ldquoflowing of timerdquo (we arenot involved in thermodynamics here) time is just one of thedimensions and on the same level as space dimensions
There is a successful recipe though
8182019 Luca Bai Otti
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We have the illusion to live in 3D and it is easier to tell computers toperform simulations (time-)step by (time-)step
Also assign a normalizationsuch that
Define therefore
So given a manifold describing a spacetime with 4-metricwe want to foliate it via space-like three-dimensionalhypersurfaces We label such hypersurfaces with the
time coordinate t
(ldquothe direction of timerdquo)
(As mostly in numer ical relativity the signature is here -+++)
The function is called the rdquolapserdquo function and itis strictly positive for spacelike hypersurfaces
α
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so that
ii) and the spatial metric
Letrsquos also define
i) the unit normal vector to the hypersurface
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The spatial part is obtained by contracting with the spatialprojection operator defined as
By using we can decompose any 4D tensor into a
purely spatial part (hence in ) and a purely timelike part (henceorthogonal to and aligned with )
while the timelike part is obtained by contracting with the timelikeprojection operator
The two projectors are obviously orthogonal
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The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariantderivative
All the 4D tensors in the Einstein equations can be projectedstraightforwardly onto the 3D spatial slice
In particular the 3D connection coefficients
the 3D Riemann tensor
and the 3D contractions of the 3D Riemann tensor ie the 3DRicci tensor the 3D Ricci scalar and
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Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic
curvatureThe extrinsic curvature is defined in terms of the unit normal to as
where is the Lie derivative along
This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric
We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic
curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold
and this can be shown to be equivalent also to
8182019 Luca Bai Otti
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Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
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Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
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W f l i EOS
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
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bull Comments
bull Einstein interpretation dispenses with any action-at-a-distance
bull Although the surface of the apple is curved if you look at any local spot closely (with amagnifying glass) its geometry looks like that of a flat surface (the Minkowski spacetime of
the special relativity)
bull The interaction of spacetime and matter is summarized in John Archibald Wheelers favorite
words spacetime tells matter how to move and matter tells spacetime how to curve
bull This reciprocal influence (matter spacetime) makes Einsteinrsquos field equation non-linearand so very hard to solve
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bull Summary of the parable
bull 1) objects follow geodesics and locally geodesics appear straight
bull 2) over more extended regions of space and time geodesics originally receding from eachother begin to approach at a rate governed by the curvature of spacetime and this effect of
geometry on matter is what was called ldquogravitationrdquo
bull 3) matter in turn warps geometry
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bull Important quantities in general relativity
bull the metric (the metric tensor g) which may be regarded as a machinery for measuring
distances
dS 2 = gmicroν dxmicrodxν
Rmicroν = Rα
microαν bull the Ricci tensor and the curvature (Ricci) scalar R = gmicroν Rmicroν
Rαβmicroν = Γ
αβν micro minus Γ
αβmicroν + Γ
ασmicroΓ
σβν minus Γ
ασν Γ
σβmicro
Γαβmicro =
12
gασ(gβσmicro + gσmicroβ minus gβmicroσ)
bull curvature expressed by the Riemann curvature tensor
(where are the Christoffel symbols)
bull covariant derivative a derivative that takes into account the curvature of the spacetime
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bull From these quantities the path of any particle can be calculated This is how geometry tells
matter how to move
bull The other direction (matter tells spacetime how to curve) requires to know the distribution of
matter (massenergymomentum) described through the stress-energy tensor T
bull After many years of thinking Einstein reached a satisfactory form for the equations relating
geometry and matter
Einstein_tensor = constant x T
bull The Einstein tensor (usually called G) is a tensor in 4D spacetime that has the wanted properties
of
bull being a symmetric tensor (it must because the stress-energy tensor is symmetric)
bull having vanishing (covariant) divergence (it must because the stress-energy tensor has
vanishing divergence)
bull the weak-field limit of the Einstein equations gives the Newtonian Poisson equation (from the
comparison to which the value of the above constant is found)
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ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
The Einstein equations
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
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bull Gravitational redshift
bull change in the frequency of light moving in regions of different curvature
bull measured in experiments and taken into account by the GPS
bull Periastron shift
bull first measured in the perihelion advance of Mercury (the GR prediction coincides with
the ldquoanomalousrdquo advance if computed in Newtonian theory)bull binary pulsar (strong fields so larger shifts)
bull Bending of light
bull first measured in the bending of photons traveling near the Sun
bull gravitational lensing
bull
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bull A black hole is literally a region from which light cannot escape
bull The boundary of a black hole is called the event horizon When something passes through the
event horizon it can no longer communicate in any way with the world outside However it is
not a concrete surface there is no local measurement that can tell an observer that heshe is on
the event horizon
bull General relativity predicts that black holes can form by gravitational collapse Once a star has
burned up its supply of nuclear fuel it can no longer support itself against its own weight and it
collapses It will reach a critical density at which an explosive re-ignition of burning occurs
called a supernova
bull There are several candidate objects that are thought to be black holes (because they are very
compact) but there has been no direct observation of black holes up to now
bull The definitive way to detect black holes is through gravitational waves (more on this later)
because gravitational waves from black holes are distinct from those of other objects
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bull In general relativity disturbances in the spacetime curvature (the ldquooldrdquo ldquogravitational
fieldrdquo) propagate at the speed of light as gravitational radiation or gravitational waves
(also known as gravity waves but this term was already in use in fluid dynamics with a
different meaning so I recommend to avoid it)
bull Gravitational waves are a strong point of general relativity which solves the action-at-a-
distance problem of Newtonian gravity
bull Analogously to light which is produced by the motion of electric charges gravitational
waves are produced by the motion of anything (mass-energy) When you wave yourhand you make gravitational waves
bull The point is that the amplitude of gravitational radiation is very small
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T R3(GM )
part 3Qij
part t3
MR2
T 3
Mv2
T
LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
LGW = 15Gc5
part 3
Qij
part t3part 3
Qij
part t3 1048575
8182019 Luca Bai Otti
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LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
G4
c5
82times
10
minus74
cgs units
c5 27 times 10minus60 cgs units
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bull In fact gravitational waves have not been detected yet even if there are strong
indications that gravitational waves exist as predicted by general relativity
bull Such an indirect evidence comes from binary pulsars Two neutron stars orbiting each
other at least one of which is a pulsar The curvature is large so it is an interesting place
to study general relativistic effects (Incidentally their orbit may precess by 4 degree per
year)bull The spin axis of pulsars is not aligned with
their magnetic axis so at each rotation theyemit pulses of radio waves and we can
detect them from the Earth Pulsars behave
as very accurate clocks
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bull Actually such measurements are very precise and
confirm that the orbit is shrinking with time at a rate
in complete agreement with the emission of
gravitational waves predicted by general relativity
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radio
far-IR
mid-IR
near-IR
opticalx-ray
gamma-ray
GWs
It has happened over and over in the history of astronomy as a newldquowindowrdquo has been opened a ldquonewrdquo universe has been revealed
The same will happen with GW-astronomy
GSFCNASA
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Large-scale
Cryogenic
Gravitational-waveTelescope
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o Binary neutron stars
o Binary black-holes
o Deformed compact stars including Low Mass X-ray Binaries (LMXB)and pulsars
o Cosmological stochastic background
o Mixed binary systems
o Gravitational collapse (supernovae neutron stars)
o Extreme-Mass-Ratio Inspirals (EMRI)
o
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TAMA-Tokyo GEO-Hannover LIGO-Livingston LIGO-Hanford VIRGO-Cascina
and LCGT will measure LL ~ h lt 10-21 with SN~1
Numericalrelativity
h
Knowledge of thewaveforms cancompensate for the
very small SN(matched-filtering) and
so enhance detection and allow for source-characterization
possible
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ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
The fundamental equations standard numerical relativity aims at solving
are the Einstein equations
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General relativity states that our World is a 4D and curvedspacetime and the Einstein equations describe its dynamics
How to solve the Einstein equations numerically
Prominently there is no a priori concept of ldquoflowing of timerdquo (we arenot involved in thermodynamics here) time is just one of thedimensions and on the same level as space dimensions
There is a successful recipe though
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We have the illusion to live in 3D and it is easier to tell computers toperform simulations (time-)step by (time-)step
Also assign a normalizationsuch that
Define therefore
So given a manifold describing a spacetime with 4-metricwe want to foliate it via space-like three-dimensionalhypersurfaces We label such hypersurfaces with the
time coordinate t
(ldquothe direction of timerdquo)
(As mostly in numer ical relativity the signature is here -+++)
The function is called the rdquolapserdquo function and itis strictly positive for spacelike hypersurfaces
α
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so that
ii) and the spatial metric
Letrsquos also define
i) the unit normal vector to the hypersurface
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The spatial part is obtained by contracting with the spatialprojection operator defined as
By using we can decompose any 4D tensor into a
purely spatial part (hence in ) and a purely timelike part (henceorthogonal to and aligned with )
while the timelike part is obtained by contracting with the timelikeprojection operator
The two projectors are obviously orthogonal
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The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariantderivative
All the 4D tensors in the Einstein equations can be projectedstraightforwardly onto the 3D spatial slice
In particular the 3D connection coefficients
the 3D Riemann tensor
and the 3D contractions of the 3D Riemann tensor ie the 3DRicci tensor the 3D Ricci scalar and
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Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic
curvatureThe extrinsic curvature is defined in terms of the unit normal to as
where is the Lie derivative along
This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric
We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic
curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold
and this can be shown to be equivalent also to
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Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
8182019 Luca Bai Otti
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 68100
Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
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W f l i EOS
8182019 Luca Bai Otti
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
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bull Summary of the parable
bull 1) objects follow geodesics and locally geodesics appear straight
bull 2) over more extended regions of space and time geodesics originally receding from eachother begin to approach at a rate governed by the curvature of spacetime and this effect of
geometry on matter is what was called ldquogravitationrdquo
bull 3) matter in turn warps geometry
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bull Important quantities in general relativity
bull the metric (the metric tensor g) which may be regarded as a machinery for measuring
distances
dS 2 = gmicroν dxmicrodxν
Rmicroν = Rα
microαν bull the Ricci tensor and the curvature (Ricci) scalar R = gmicroν Rmicroν
Rαβmicroν = Γ
αβν micro minus Γ
αβmicroν + Γ
ασmicroΓ
σβν minus Γ
ασν Γ
σβmicro
Γαβmicro =
12
gασ(gβσmicro + gσmicroβ minus gβmicroσ)
bull curvature expressed by the Riemann curvature tensor
(where are the Christoffel symbols)
bull covariant derivative a derivative that takes into account the curvature of the spacetime
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bull From these quantities the path of any particle can be calculated This is how geometry tells
matter how to move
bull The other direction (matter tells spacetime how to curve) requires to know the distribution of
matter (massenergymomentum) described through the stress-energy tensor T
bull After many years of thinking Einstein reached a satisfactory form for the equations relating
geometry and matter
Einstein_tensor = constant x T
bull The Einstein tensor (usually called G) is a tensor in 4D spacetime that has the wanted properties
of
bull being a symmetric tensor (it must because the stress-energy tensor is symmetric)
bull having vanishing (covariant) divergence (it must because the stress-energy tensor has
vanishing divergence)
bull the weak-field limit of the Einstein equations gives the Newtonian Poisson equation (from the
comparison to which the value of the above constant is found)
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ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
The Einstein equations
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
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bull Gravitational redshift
bull change in the frequency of light moving in regions of different curvature
bull measured in experiments and taken into account by the GPS
bull Periastron shift
bull first measured in the perihelion advance of Mercury (the GR prediction coincides with
the ldquoanomalousrdquo advance if computed in Newtonian theory)bull binary pulsar (strong fields so larger shifts)
bull Bending of light
bull first measured in the bending of photons traveling near the Sun
bull gravitational lensing
bull
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bull A black hole is literally a region from which light cannot escape
bull The boundary of a black hole is called the event horizon When something passes through the
event horizon it can no longer communicate in any way with the world outside However it is
not a concrete surface there is no local measurement that can tell an observer that heshe is on
the event horizon
bull General relativity predicts that black holes can form by gravitational collapse Once a star has
burned up its supply of nuclear fuel it can no longer support itself against its own weight and it
collapses It will reach a critical density at which an explosive re-ignition of burning occurs
called a supernova
bull There are several candidate objects that are thought to be black holes (because they are very
compact) but there has been no direct observation of black holes up to now
bull The definitive way to detect black holes is through gravitational waves (more on this later)
because gravitational waves from black holes are distinct from those of other objects
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bull In general relativity disturbances in the spacetime curvature (the ldquooldrdquo ldquogravitational
fieldrdquo) propagate at the speed of light as gravitational radiation or gravitational waves
(also known as gravity waves but this term was already in use in fluid dynamics with a
different meaning so I recommend to avoid it)
bull Gravitational waves are a strong point of general relativity which solves the action-at-a-
distance problem of Newtonian gravity
bull Analogously to light which is produced by the motion of electric charges gravitational
waves are produced by the motion of anything (mass-energy) When you wave yourhand you make gravitational waves
bull The point is that the amplitude of gravitational radiation is very small
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T R3(GM )
part 3Qij
part t3
MR2
T 3
Mv2
T
LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
LGW = 15Gc5
part 3
Qij
part t3part 3
Qij
part t3 1048575
8182019 Luca Bai Otti
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LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
G4
c5
82times
10
minus74
cgs units
c5 27 times 10minus60 cgs units
8182019 Luca Bai Otti
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bull In fact gravitational waves have not been detected yet even if there are strong
indications that gravitational waves exist as predicted by general relativity
bull Such an indirect evidence comes from binary pulsars Two neutron stars orbiting each
other at least one of which is a pulsar The curvature is large so it is an interesting place
to study general relativistic effects (Incidentally their orbit may precess by 4 degree per
year)bull The spin axis of pulsars is not aligned with
their magnetic axis so at each rotation theyemit pulses of radio waves and we can
detect them from the Earth Pulsars behave
as very accurate clocks
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bull Actually such measurements are very precise and
confirm that the orbit is shrinking with time at a rate
in complete agreement with the emission of
gravitational waves predicted by general relativity
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radio
far-IR
mid-IR
near-IR
opticalx-ray
gamma-ray
GWs
It has happened over and over in the history of astronomy as a newldquowindowrdquo has been opened a ldquonewrdquo universe has been revealed
The same will happen with GW-astronomy
GSFCNASA
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Large-scale
Cryogenic
Gravitational-waveTelescope
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o Binary neutron stars
o Binary black-holes
o Deformed compact stars including Low Mass X-ray Binaries (LMXB)and pulsars
o Cosmological stochastic background
o Mixed binary systems
o Gravitational collapse (supernovae neutron stars)
o Extreme-Mass-Ratio Inspirals (EMRI)
o
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8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 28100
TAMA-Tokyo GEO-Hannover LIGO-Livingston LIGO-Hanford VIRGO-Cascina
and LCGT will measure LL ~ h lt 10-21 with SN~1
Numericalrelativity
h
Knowledge of thewaveforms cancompensate for the
very small SN(matched-filtering) and
so enhance detection and allow for source-characterization
possible
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httpslidepdfcomreaderfullluca-bai-otti 29100
ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
The fundamental equations standard numerical relativity aims at solving
are the Einstein equations
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General relativity states that our World is a 4D and curvedspacetime and the Einstein equations describe its dynamics
How to solve the Einstein equations numerically
Prominently there is no a priori concept of ldquoflowing of timerdquo (we arenot involved in thermodynamics here) time is just one of thedimensions and on the same level as space dimensions
There is a successful recipe though
8182019 Luca Bai Otti
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We have the illusion to live in 3D and it is easier to tell computers toperform simulations (time-)step by (time-)step
Also assign a normalizationsuch that
Define therefore
So given a manifold describing a spacetime with 4-metricwe want to foliate it via space-like three-dimensionalhypersurfaces We label such hypersurfaces with the
time coordinate t
(ldquothe direction of timerdquo)
(As mostly in numer ical relativity the signature is here -+++)
The function is called the rdquolapserdquo function and itis strictly positive for spacelike hypersurfaces
α
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so that
ii) and the spatial metric
Letrsquos also define
i) the unit normal vector to the hypersurface
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The spatial part is obtained by contracting with the spatialprojection operator defined as
By using we can decompose any 4D tensor into a
purely spatial part (hence in ) and a purely timelike part (henceorthogonal to and aligned with )
while the timelike part is obtained by contracting with the timelikeprojection operator
The two projectors are obviously orthogonal
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The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariantderivative
All the 4D tensors in the Einstein equations can be projectedstraightforwardly onto the 3D spatial slice
In particular the 3D connection coefficients
the 3D Riemann tensor
and the 3D contractions of the 3D Riemann tensor ie the 3DRicci tensor the 3D Ricci scalar and
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Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic
curvatureThe extrinsic curvature is defined in terms of the unit normal to as
where is the Lie derivative along
This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric
We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic
curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold
and this can be shown to be equivalent also to
8182019 Luca Bai Otti
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Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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httpslidepdfcomreaderfullluca-bai-otti 38100
Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
8182019 Luca Bai Otti
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
8182019 Luca Bai Otti
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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httpslidepdfcomreaderfullluca-bai-otti 48100
As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
8182019 Luca Bai Otti
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
8182019 Luca Bai Otti
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
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Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
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M d i
8182019 Luca Bai Otti
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
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W f l i EOS
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
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bull Important quantities in general relativity
bull the metric (the metric tensor g) which may be regarded as a machinery for measuring
distances
dS 2 = gmicroν dxmicrodxν
Rmicroν = Rα
microαν bull the Ricci tensor and the curvature (Ricci) scalar R = gmicroν Rmicroν
Rαβmicroν = Γ
αβν micro minus Γ
αβmicroν + Γ
ασmicroΓ
σβν minus Γ
ασν Γ
σβmicro
Γαβmicro =
12
gασ(gβσmicro + gσmicroβ minus gβmicroσ)
bull curvature expressed by the Riemann curvature tensor
(where are the Christoffel symbols)
bull covariant derivative a derivative that takes into account the curvature of the spacetime
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bull From these quantities the path of any particle can be calculated This is how geometry tells
matter how to move
bull The other direction (matter tells spacetime how to curve) requires to know the distribution of
matter (massenergymomentum) described through the stress-energy tensor T
bull After many years of thinking Einstein reached a satisfactory form for the equations relating
geometry and matter
Einstein_tensor = constant x T
bull The Einstein tensor (usually called G) is a tensor in 4D spacetime that has the wanted properties
of
bull being a symmetric tensor (it must because the stress-energy tensor is symmetric)
bull having vanishing (covariant) divergence (it must because the stress-energy tensor has
vanishing divergence)
bull the weak-field limit of the Einstein equations gives the Newtonian Poisson equation (from the
comparison to which the value of the above constant is found)
8182019 Luca Bai Otti
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ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
The Einstein equations
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
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httpslidepdfcomreaderfullluca-bai-otti 15100
bull Gravitational redshift
bull change in the frequency of light moving in regions of different curvature
bull measured in experiments and taken into account by the GPS
bull Periastron shift
bull first measured in the perihelion advance of Mercury (the GR prediction coincides with
the ldquoanomalousrdquo advance if computed in Newtonian theory)bull binary pulsar (strong fields so larger shifts)
bull Bending of light
bull first measured in the bending of photons traveling near the Sun
bull gravitational lensing
bull
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bull A black hole is literally a region from which light cannot escape
bull The boundary of a black hole is called the event horizon When something passes through the
event horizon it can no longer communicate in any way with the world outside However it is
not a concrete surface there is no local measurement that can tell an observer that heshe is on
the event horizon
bull General relativity predicts that black holes can form by gravitational collapse Once a star has
burned up its supply of nuclear fuel it can no longer support itself against its own weight and it
collapses It will reach a critical density at which an explosive re-ignition of burning occurs
called a supernova
bull There are several candidate objects that are thought to be black holes (because they are very
compact) but there has been no direct observation of black holes up to now
bull The definitive way to detect black holes is through gravitational waves (more on this later)
because gravitational waves from black holes are distinct from those of other objects
8182019 Luca Bai Otti
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bull In general relativity disturbances in the spacetime curvature (the ldquooldrdquo ldquogravitational
fieldrdquo) propagate at the speed of light as gravitational radiation or gravitational waves
(also known as gravity waves but this term was already in use in fluid dynamics with a
different meaning so I recommend to avoid it)
bull Gravitational waves are a strong point of general relativity which solves the action-at-a-
distance problem of Newtonian gravity
bull Analogously to light which is produced by the motion of electric charges gravitational
waves are produced by the motion of anything (mass-energy) When you wave yourhand you make gravitational waves
bull The point is that the amplitude of gravitational radiation is very small
8182019 Luca Bai Otti
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T R3(GM )
part 3Qij
part t3
MR2
T 3
Mv2
T
LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
LGW = 15Gc5
part 3
Qij
part t3part 3
Qij
part t3 1048575
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 19100
LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
G4
c5
82times
10
minus74
cgs units
c5 27 times 10minus60 cgs units
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 20100
bull In fact gravitational waves have not been detected yet even if there are strong
indications that gravitational waves exist as predicted by general relativity
bull Such an indirect evidence comes from binary pulsars Two neutron stars orbiting each
other at least one of which is a pulsar The curvature is large so it is an interesting place
to study general relativistic effects (Incidentally their orbit may precess by 4 degree per
year)bull The spin axis of pulsars is not aligned with
their magnetic axis so at each rotation theyemit pulses of radio waves and we can
detect them from the Earth Pulsars behave
as very accurate clocks
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bull Actually such measurements are very precise and
confirm that the orbit is shrinking with time at a rate
in complete agreement with the emission of
gravitational waves predicted by general relativity
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radio
far-IR
mid-IR
near-IR
opticalx-ray
gamma-ray
GWs
It has happened over and over in the history of astronomy as a newldquowindowrdquo has been opened a ldquonewrdquo universe has been revealed
The same will happen with GW-astronomy
GSFCNASA
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httpslidepdfcomreaderfullluca-bai-otti 24100
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Large-scale
Cryogenic
Gravitational-waveTelescope
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o Binary neutron stars
o Binary black-holes
o Deformed compact stars including Low Mass X-ray Binaries (LMXB)and pulsars
o Cosmological stochastic background
o Mixed binary systems
o Gravitational collapse (supernovae neutron stars)
o Extreme-Mass-Ratio Inspirals (EMRI)
o
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TAMA-Tokyo GEO-Hannover LIGO-Livingston LIGO-Hanford VIRGO-Cascina
and LCGT will measure LL ~ h lt 10-21 with SN~1
Numericalrelativity
h
Knowledge of thewaveforms cancompensate for the
very small SN(matched-filtering) and
so enhance detection and allow for source-characterization
possible
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 29100
ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
The fundamental equations standard numerical relativity aims at solving
are the Einstein equations
8182019 Luca Bai Otti
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General relativity states that our World is a 4D and curvedspacetime and the Einstein equations describe its dynamics
How to solve the Einstein equations numerically
Prominently there is no a priori concept of ldquoflowing of timerdquo (we arenot involved in thermodynamics here) time is just one of thedimensions and on the same level as space dimensions
There is a successful recipe though
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We have the illusion to live in 3D and it is easier to tell computers toperform simulations (time-)step by (time-)step
Also assign a normalizationsuch that
Define therefore
So given a manifold describing a spacetime with 4-metricwe want to foliate it via space-like three-dimensionalhypersurfaces We label such hypersurfaces with the
time coordinate t
(ldquothe direction of timerdquo)
(As mostly in numer ical relativity the signature is here -+++)
The function is called the rdquolapserdquo function and itis strictly positive for spacelike hypersurfaces
α
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so that
ii) and the spatial metric
Letrsquos also define
i) the unit normal vector to the hypersurface
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The spatial part is obtained by contracting with the spatialprojection operator defined as
By using we can decompose any 4D tensor into a
purely spatial part (hence in ) and a purely timelike part (henceorthogonal to and aligned with )
while the timelike part is obtained by contracting with the timelikeprojection operator
The two projectors are obviously orthogonal
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The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariantderivative
All the 4D tensors in the Einstein equations can be projectedstraightforwardly onto the 3D spatial slice
In particular the 3D connection coefficients
the 3D Riemann tensor
and the 3D contractions of the 3D Riemann tensor ie the 3DRicci tensor the 3D Ricci scalar and
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Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic
curvatureThe extrinsic curvature is defined in terms of the unit normal to as
where is the Lie derivative along
This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric
We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic
curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold
and this can be shown to be equivalent also to
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Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
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Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
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W f l i EOS
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
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bull From these quantities the path of any particle can be calculated This is how geometry tells
matter how to move
bull The other direction (matter tells spacetime how to curve) requires to know the distribution of
matter (massenergymomentum) described through the stress-energy tensor T
bull After many years of thinking Einstein reached a satisfactory form for the equations relating
geometry and matter
Einstein_tensor = constant x T
bull The Einstein tensor (usually called G) is a tensor in 4D spacetime that has the wanted properties
of
bull being a symmetric tensor (it must because the stress-energy tensor is symmetric)
bull having vanishing (covariant) divergence (it must because the stress-energy tensor has
vanishing divergence)
bull the weak-field limit of the Einstein equations gives the Newtonian Poisson equation (from the
comparison to which the value of the above constant is found)
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ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
The Einstein equations
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
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bull Gravitational redshift
bull change in the frequency of light moving in regions of different curvature
bull measured in experiments and taken into account by the GPS
bull Periastron shift
bull first measured in the perihelion advance of Mercury (the GR prediction coincides with
the ldquoanomalousrdquo advance if computed in Newtonian theory)bull binary pulsar (strong fields so larger shifts)
bull Bending of light
bull first measured in the bending of photons traveling near the Sun
bull gravitational lensing
bull
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bull A black hole is literally a region from which light cannot escape
bull The boundary of a black hole is called the event horizon When something passes through the
event horizon it can no longer communicate in any way with the world outside However it is
not a concrete surface there is no local measurement that can tell an observer that heshe is on
the event horizon
bull General relativity predicts that black holes can form by gravitational collapse Once a star has
burned up its supply of nuclear fuel it can no longer support itself against its own weight and it
collapses It will reach a critical density at which an explosive re-ignition of burning occurs
called a supernova
bull There are several candidate objects that are thought to be black holes (because they are very
compact) but there has been no direct observation of black holes up to now
bull The definitive way to detect black holes is through gravitational waves (more on this later)
because gravitational waves from black holes are distinct from those of other objects
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bull In general relativity disturbances in the spacetime curvature (the ldquooldrdquo ldquogravitational
fieldrdquo) propagate at the speed of light as gravitational radiation or gravitational waves
(also known as gravity waves but this term was already in use in fluid dynamics with a
different meaning so I recommend to avoid it)
bull Gravitational waves are a strong point of general relativity which solves the action-at-a-
distance problem of Newtonian gravity
bull Analogously to light which is produced by the motion of electric charges gravitational
waves are produced by the motion of anything (mass-energy) When you wave yourhand you make gravitational waves
bull The point is that the amplitude of gravitational radiation is very small
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T R3(GM )
part 3Qij
part t3
MR2
T 3
Mv2
T
LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
LGW = 15Gc5
part 3
Qij
part t3part 3
Qij
part t3 1048575
8182019 Luca Bai Otti
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LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
G4
c5
82times
10
minus74
cgs units
c5 27 times 10minus60 cgs units
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bull In fact gravitational waves have not been detected yet even if there are strong
indications that gravitational waves exist as predicted by general relativity
bull Such an indirect evidence comes from binary pulsars Two neutron stars orbiting each
other at least one of which is a pulsar The curvature is large so it is an interesting place
to study general relativistic effects (Incidentally their orbit may precess by 4 degree per
year)bull The spin axis of pulsars is not aligned with
their magnetic axis so at each rotation theyemit pulses of radio waves and we can
detect them from the Earth Pulsars behave
as very accurate clocks
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bull Actually such measurements are very precise and
confirm that the orbit is shrinking with time at a rate
in complete agreement with the emission of
gravitational waves predicted by general relativity
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radio
far-IR
mid-IR
near-IR
opticalx-ray
gamma-ray
GWs
It has happened over and over in the history of astronomy as a newldquowindowrdquo has been opened a ldquonewrdquo universe has been revealed
The same will happen with GW-astronomy
GSFCNASA
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Large-scale
Cryogenic
Gravitational-waveTelescope
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o Binary neutron stars
o Binary black-holes
o Deformed compact stars including Low Mass X-ray Binaries (LMXB)and pulsars
o Cosmological stochastic background
o Mixed binary systems
o Gravitational collapse (supernovae neutron stars)
o Extreme-Mass-Ratio Inspirals (EMRI)
o
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TAMA-Tokyo GEO-Hannover LIGO-Livingston LIGO-Hanford VIRGO-Cascina
and LCGT will measure LL ~ h lt 10-21 with SN~1
Numericalrelativity
h
Knowledge of thewaveforms cancompensate for the
very small SN(matched-filtering) and
so enhance detection and allow for source-characterization
possible
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ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
The fundamental equations standard numerical relativity aims at solving
are the Einstein equations
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General relativity states that our World is a 4D and curvedspacetime and the Einstein equations describe its dynamics
How to solve the Einstein equations numerically
Prominently there is no a priori concept of ldquoflowing of timerdquo (we arenot involved in thermodynamics here) time is just one of thedimensions and on the same level as space dimensions
There is a successful recipe though
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We have the illusion to live in 3D and it is easier to tell computers toperform simulations (time-)step by (time-)step
Also assign a normalizationsuch that
Define therefore
So given a manifold describing a spacetime with 4-metricwe want to foliate it via space-like three-dimensionalhypersurfaces We label such hypersurfaces with the
time coordinate t
(ldquothe direction of timerdquo)
(As mostly in numer ical relativity the signature is here -+++)
The function is called the rdquolapserdquo function and itis strictly positive for spacelike hypersurfaces
α
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so that
ii) and the spatial metric
Letrsquos also define
i) the unit normal vector to the hypersurface
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The spatial part is obtained by contracting with the spatialprojection operator defined as
By using we can decompose any 4D tensor into a
purely spatial part (hence in ) and a purely timelike part (henceorthogonal to and aligned with )
while the timelike part is obtained by contracting with the timelikeprojection operator
The two projectors are obviously orthogonal
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The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariantderivative
All the 4D tensors in the Einstein equations can be projectedstraightforwardly onto the 3D spatial slice
In particular the 3D connection coefficients
the 3D Riemann tensor
and the 3D contractions of the 3D Riemann tensor ie the 3DRicci tensor the 3D Ricci scalar and
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Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic
curvatureThe extrinsic curvature is defined in terms of the unit normal to as
where is the Lie derivative along
This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric
We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic
curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold
and this can be shown to be equivalent also to
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Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
8182019 Luca Bai Otti
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
8182019 Luca Bai Otti
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
8182019 Luca Bai Otti
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
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Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
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W f l i EOS
8182019 Luca Bai Otti
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
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ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
The Einstein equations
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
8182019 Luca Bai Otti
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bull Gravitational redshift
bull change in the frequency of light moving in regions of different curvature
bull measured in experiments and taken into account by the GPS
bull Periastron shift
bull first measured in the perihelion advance of Mercury (the GR prediction coincides with
the ldquoanomalousrdquo advance if computed in Newtonian theory)bull binary pulsar (strong fields so larger shifts)
bull Bending of light
bull first measured in the bending of photons traveling near the Sun
bull gravitational lensing
bull
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bull A black hole is literally a region from which light cannot escape
bull The boundary of a black hole is called the event horizon When something passes through the
event horizon it can no longer communicate in any way with the world outside However it is
not a concrete surface there is no local measurement that can tell an observer that heshe is on
the event horizon
bull General relativity predicts that black holes can form by gravitational collapse Once a star has
burned up its supply of nuclear fuel it can no longer support itself against its own weight and it
collapses It will reach a critical density at which an explosive re-ignition of burning occurs
called a supernova
bull There are several candidate objects that are thought to be black holes (because they are very
compact) but there has been no direct observation of black holes up to now
bull The definitive way to detect black holes is through gravitational waves (more on this later)
because gravitational waves from black holes are distinct from those of other objects
8182019 Luca Bai Otti
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bull In general relativity disturbances in the spacetime curvature (the ldquooldrdquo ldquogravitational
fieldrdquo) propagate at the speed of light as gravitational radiation or gravitational waves
(also known as gravity waves but this term was already in use in fluid dynamics with a
different meaning so I recommend to avoid it)
bull Gravitational waves are a strong point of general relativity which solves the action-at-a-
distance problem of Newtonian gravity
bull Analogously to light which is produced by the motion of electric charges gravitational
waves are produced by the motion of anything (mass-energy) When you wave yourhand you make gravitational waves
bull The point is that the amplitude of gravitational radiation is very small
8182019 Luca Bai Otti
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T R3(GM )
part 3Qij
part t3
MR2
T 3
Mv2
T
LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
LGW = 15Gc5
part 3
Qij
part t3part 3
Qij
part t3 1048575
8182019 Luca Bai Otti
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LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
G4
c5
82times
10
minus74
cgs units
c5 27 times 10minus60 cgs units
8182019 Luca Bai Otti
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bull In fact gravitational waves have not been detected yet even if there are strong
indications that gravitational waves exist as predicted by general relativity
bull Such an indirect evidence comes from binary pulsars Two neutron stars orbiting each
other at least one of which is a pulsar The curvature is large so it is an interesting place
to study general relativistic effects (Incidentally their orbit may precess by 4 degree per
year)bull The spin axis of pulsars is not aligned with
their magnetic axis so at each rotation theyemit pulses of radio waves and we can
detect them from the Earth Pulsars behave
as very accurate clocks
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bull Actually such measurements are very precise and
confirm that the orbit is shrinking with time at a rate
in complete agreement with the emission of
gravitational waves predicted by general relativity
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radio
far-IR
mid-IR
near-IR
opticalx-ray
gamma-ray
GWs
It has happened over and over in the history of astronomy as a newldquowindowrdquo has been opened a ldquonewrdquo universe has been revealed
The same will happen with GW-astronomy
GSFCNASA
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Large-scale
Cryogenic
Gravitational-waveTelescope
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o Binary neutron stars
o Binary black-holes
o Deformed compact stars including Low Mass X-ray Binaries (LMXB)and pulsars
o Cosmological stochastic background
o Mixed binary systems
o Gravitational collapse (supernovae neutron stars)
o Extreme-Mass-Ratio Inspirals (EMRI)
o
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TAMA-Tokyo GEO-Hannover LIGO-Livingston LIGO-Hanford VIRGO-Cascina
and LCGT will measure LL ~ h lt 10-21 with SN~1
Numericalrelativity
h
Knowledge of thewaveforms cancompensate for the
very small SN(matched-filtering) and
so enhance detection and allow for source-characterization
possible
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httpslidepdfcomreaderfullluca-bai-otti 29100
ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
The fundamental equations standard numerical relativity aims at solving
are the Einstein equations
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General relativity states that our World is a 4D and curvedspacetime and the Einstein equations describe its dynamics
How to solve the Einstein equations numerically
Prominently there is no a priori concept of ldquoflowing of timerdquo (we arenot involved in thermodynamics here) time is just one of thedimensions and on the same level as space dimensions
There is a successful recipe though
8182019 Luca Bai Otti
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We have the illusion to live in 3D and it is easier to tell computers toperform simulations (time-)step by (time-)step
Also assign a normalizationsuch that
Define therefore
So given a manifold describing a spacetime with 4-metricwe want to foliate it via space-like three-dimensionalhypersurfaces We label such hypersurfaces with the
time coordinate t
(ldquothe direction of timerdquo)
(As mostly in numer ical relativity the signature is here -+++)
The function is called the rdquolapserdquo function and itis strictly positive for spacelike hypersurfaces
α
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so that
ii) and the spatial metric
Letrsquos also define
i) the unit normal vector to the hypersurface
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The spatial part is obtained by contracting with the spatialprojection operator defined as
By using we can decompose any 4D tensor into a
purely spatial part (hence in ) and a purely timelike part (henceorthogonal to and aligned with )
while the timelike part is obtained by contracting with the timelikeprojection operator
The two projectors are obviously orthogonal
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The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariantderivative
All the 4D tensors in the Einstein equations can be projectedstraightforwardly onto the 3D spatial slice
In particular the 3D connection coefficients
the 3D Riemann tensor
and the 3D contractions of the 3D Riemann tensor ie the 3DRicci tensor the 3D Ricci scalar and
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Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic
curvatureThe extrinsic curvature is defined in terms of the unit normal to as
where is the Lie derivative along
This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric
We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic
curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold
and this can be shown to be equivalent also to
8182019 Luca Bai Otti
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Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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httpslidepdfcomreaderfullluca-bai-otti 38100
Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
8182019 Luca Bai Otti
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
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Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
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W f l i EOS
8182019 Luca Bai Otti
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
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httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
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bull Gravitational redshift
bull change in the frequency of light moving in regions of different curvature
bull measured in experiments and taken into account by the GPS
bull Periastron shift
bull first measured in the perihelion advance of Mercury (the GR prediction coincides with
the ldquoanomalousrdquo advance if computed in Newtonian theory)bull binary pulsar (strong fields so larger shifts)
bull Bending of light
bull first measured in the bending of photons traveling near the Sun
bull gravitational lensing
bull
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bull A black hole is literally a region from which light cannot escape
bull The boundary of a black hole is called the event horizon When something passes through the
event horizon it can no longer communicate in any way with the world outside However it is
not a concrete surface there is no local measurement that can tell an observer that heshe is on
the event horizon
bull General relativity predicts that black holes can form by gravitational collapse Once a star has
burned up its supply of nuclear fuel it can no longer support itself against its own weight and it
collapses It will reach a critical density at which an explosive re-ignition of burning occurs
called a supernova
bull There are several candidate objects that are thought to be black holes (because they are very
compact) but there has been no direct observation of black holes up to now
bull The definitive way to detect black holes is through gravitational waves (more on this later)
because gravitational waves from black holes are distinct from those of other objects
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bull In general relativity disturbances in the spacetime curvature (the ldquooldrdquo ldquogravitational
fieldrdquo) propagate at the speed of light as gravitational radiation or gravitational waves
(also known as gravity waves but this term was already in use in fluid dynamics with a
different meaning so I recommend to avoid it)
bull Gravitational waves are a strong point of general relativity which solves the action-at-a-
distance problem of Newtonian gravity
bull Analogously to light which is produced by the motion of electric charges gravitational
waves are produced by the motion of anything (mass-energy) When you wave yourhand you make gravitational waves
bull The point is that the amplitude of gravitational radiation is very small
8182019 Luca Bai Otti
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T R3(GM )
part 3Qij
part t3
MR2
T 3
Mv2
T
LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
LGW = 15Gc5
part 3
Qij
part t3part 3
Qij
part t3 1048575
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 19100
LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
G4
c5
82times
10
minus74
cgs units
c5 27 times 10minus60 cgs units
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 20100
bull In fact gravitational waves have not been detected yet even if there are strong
indications that gravitational waves exist as predicted by general relativity
bull Such an indirect evidence comes from binary pulsars Two neutron stars orbiting each
other at least one of which is a pulsar The curvature is large so it is an interesting place
to study general relativistic effects (Incidentally their orbit may precess by 4 degree per
year)bull The spin axis of pulsars is not aligned with
their magnetic axis so at each rotation theyemit pulses of radio waves and we can
detect them from the Earth Pulsars behave
as very accurate clocks
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bull Actually such measurements are very precise and
confirm that the orbit is shrinking with time at a rate
in complete agreement with the emission of
gravitational waves predicted by general relativity
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radio
far-IR
mid-IR
near-IR
opticalx-ray
gamma-ray
GWs
It has happened over and over in the history of astronomy as a newldquowindowrdquo has been opened a ldquonewrdquo universe has been revealed
The same will happen with GW-astronomy
GSFCNASA
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Large-scale
Cryogenic
Gravitational-waveTelescope
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o Binary neutron stars
o Binary black-holes
o Deformed compact stars including Low Mass X-ray Binaries (LMXB)and pulsars
o Cosmological stochastic background
o Mixed binary systems
o Gravitational collapse (supernovae neutron stars)
o Extreme-Mass-Ratio Inspirals (EMRI)
o
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TAMA-Tokyo GEO-Hannover LIGO-Livingston LIGO-Hanford VIRGO-Cascina
and LCGT will measure LL ~ h lt 10-21 with SN~1
Numericalrelativity
h
Knowledge of thewaveforms cancompensate for the
very small SN(matched-filtering) and
so enhance detection and allow for source-characterization
possible
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ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
The fundamental equations standard numerical relativity aims at solving
are the Einstein equations
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General relativity states that our World is a 4D and curvedspacetime and the Einstein equations describe its dynamics
How to solve the Einstein equations numerically
Prominently there is no a priori concept of ldquoflowing of timerdquo (we arenot involved in thermodynamics here) time is just one of thedimensions and on the same level as space dimensions
There is a successful recipe though
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We have the illusion to live in 3D and it is easier to tell computers toperform simulations (time-)step by (time-)step
Also assign a normalizationsuch that
Define therefore
So given a manifold describing a spacetime with 4-metricwe want to foliate it via space-like three-dimensionalhypersurfaces We label such hypersurfaces with the
time coordinate t
(ldquothe direction of timerdquo)
(As mostly in numer ical relativity the signature is here -+++)
The function is called the rdquolapserdquo function and itis strictly positive for spacelike hypersurfaces
α
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so that
ii) and the spatial metric
Letrsquos also define
i) the unit normal vector to the hypersurface
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The spatial part is obtained by contracting with the spatialprojection operator defined as
By using we can decompose any 4D tensor into a
purely spatial part (hence in ) and a purely timelike part (henceorthogonal to and aligned with )
while the timelike part is obtained by contracting with the timelikeprojection operator
The two projectors are obviously orthogonal
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The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariantderivative
All the 4D tensors in the Einstein equations can be projectedstraightforwardly onto the 3D spatial slice
In particular the 3D connection coefficients
the 3D Riemann tensor
and the 3D contractions of the 3D Riemann tensor ie the 3DRicci tensor the 3D Ricci scalar and
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Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic
curvatureThe extrinsic curvature is defined in terms of the unit normal to as
where is the Lie derivative along
This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric
We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic
curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold
and this can be shown to be equivalent also to
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Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
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Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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8182019 Luca Bai Otti
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
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W f l i EOS
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
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bull A black hole is literally a region from which light cannot escape
bull The boundary of a black hole is called the event horizon When something passes through the
event horizon it can no longer communicate in any way with the world outside However it is
not a concrete surface there is no local measurement that can tell an observer that heshe is on
the event horizon
bull General relativity predicts that black holes can form by gravitational collapse Once a star has
burned up its supply of nuclear fuel it can no longer support itself against its own weight and it
collapses It will reach a critical density at which an explosive re-ignition of burning occurs
called a supernova
bull There are several candidate objects that are thought to be black holes (because they are very
compact) but there has been no direct observation of black holes up to now
bull The definitive way to detect black holes is through gravitational waves (more on this later)
because gravitational waves from black holes are distinct from those of other objects
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bull In general relativity disturbances in the spacetime curvature (the ldquooldrdquo ldquogravitational
fieldrdquo) propagate at the speed of light as gravitational radiation or gravitational waves
(also known as gravity waves but this term was already in use in fluid dynamics with a
different meaning so I recommend to avoid it)
bull Gravitational waves are a strong point of general relativity which solves the action-at-a-
distance problem of Newtonian gravity
bull Analogously to light which is produced by the motion of electric charges gravitational
waves are produced by the motion of anything (mass-energy) When you wave yourhand you make gravitational waves
bull The point is that the amplitude of gravitational radiation is very small
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T R3(GM )
part 3Qij
part t3
MR2
T 3
Mv2
T
LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
LGW = 15Gc5
part 3
Qij
part t3part 3
Qij
part t3 1048575
8182019 Luca Bai Otti
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LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
G4
c5
82times
10
minus74
cgs units
c5 27 times 10minus60 cgs units
8182019 Luca Bai Otti
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bull In fact gravitational waves have not been detected yet even if there are strong
indications that gravitational waves exist as predicted by general relativity
bull Such an indirect evidence comes from binary pulsars Two neutron stars orbiting each
other at least one of which is a pulsar The curvature is large so it is an interesting place
to study general relativistic effects (Incidentally their orbit may precess by 4 degree per
year)bull The spin axis of pulsars is not aligned with
their magnetic axis so at each rotation theyemit pulses of radio waves and we can
detect them from the Earth Pulsars behave
as very accurate clocks
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bull Actually such measurements are very precise and
confirm that the orbit is shrinking with time at a rate
in complete agreement with the emission of
gravitational waves predicted by general relativity
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radio
far-IR
mid-IR
near-IR
opticalx-ray
gamma-ray
GWs
It has happened over and over in the history of astronomy as a newldquowindowrdquo has been opened a ldquonewrdquo universe has been revealed
The same will happen with GW-astronomy
GSFCNASA
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Large-scale
Cryogenic
Gravitational-waveTelescope
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o Binary neutron stars
o Binary black-holes
o Deformed compact stars including Low Mass X-ray Binaries (LMXB)and pulsars
o Cosmological stochastic background
o Mixed binary systems
o Gravitational collapse (supernovae neutron stars)
o Extreme-Mass-Ratio Inspirals (EMRI)
o
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TAMA-Tokyo GEO-Hannover LIGO-Livingston LIGO-Hanford VIRGO-Cascina
and LCGT will measure LL ~ h lt 10-21 with SN~1
Numericalrelativity
h
Knowledge of thewaveforms cancompensate for the
very small SN(matched-filtering) and
so enhance detection and allow for source-characterization
possible
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ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
The fundamental equations standard numerical relativity aims at solving
are the Einstein equations
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General relativity states that our World is a 4D and curvedspacetime and the Einstein equations describe its dynamics
How to solve the Einstein equations numerically
Prominently there is no a priori concept of ldquoflowing of timerdquo (we arenot involved in thermodynamics here) time is just one of thedimensions and on the same level as space dimensions
There is a successful recipe though
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We have the illusion to live in 3D and it is easier to tell computers toperform simulations (time-)step by (time-)step
Also assign a normalizationsuch that
Define therefore
So given a manifold describing a spacetime with 4-metricwe want to foliate it via space-like three-dimensionalhypersurfaces We label such hypersurfaces with the
time coordinate t
(ldquothe direction of timerdquo)
(As mostly in numer ical relativity the signature is here -+++)
The function is called the rdquolapserdquo function and itis strictly positive for spacelike hypersurfaces
α
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so that
ii) and the spatial metric
Letrsquos also define
i) the unit normal vector to the hypersurface
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The spatial part is obtained by contracting with the spatialprojection operator defined as
By using we can decompose any 4D tensor into a
purely spatial part (hence in ) and a purely timelike part (henceorthogonal to and aligned with )
while the timelike part is obtained by contracting with the timelikeprojection operator
The two projectors are obviously orthogonal
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The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariantderivative
All the 4D tensors in the Einstein equations can be projectedstraightforwardly onto the 3D spatial slice
In particular the 3D connection coefficients
the 3D Riemann tensor
and the 3D contractions of the 3D Riemann tensor ie the 3DRicci tensor the 3D Ricci scalar and
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Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic
curvatureThe extrinsic curvature is defined in terms of the unit normal to as
where is the Lie derivative along
This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric
We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic
curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold
and this can be shown to be equivalent also to
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Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
8182019 Luca Bai Otti
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
8182019 Luca Bai Otti
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
8182019 Luca Bai Otti
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 68100
Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
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W f l i EOS
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
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bull In general relativity disturbances in the spacetime curvature (the ldquooldrdquo ldquogravitational
fieldrdquo) propagate at the speed of light as gravitational radiation or gravitational waves
(also known as gravity waves but this term was already in use in fluid dynamics with a
different meaning so I recommend to avoid it)
bull Gravitational waves are a strong point of general relativity which solves the action-at-a-
distance problem of Newtonian gravity
bull Analogously to light which is produced by the motion of electric charges gravitational
waves are produced by the motion of anything (mass-energy) When you wave yourhand you make gravitational waves
bull The point is that the amplitude of gravitational radiation is very small
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T R3(GM )
part 3Qij
part t3
MR2
T 3
Mv2
T
LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
LGW = 15Gc5
part 3
Qij
part t3part 3
Qij
part t3 1048575
8182019 Luca Bai Otti
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LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
G4
c5
82times
10
minus74
cgs units
c5 27 times 10minus60 cgs units
8182019 Luca Bai Otti
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bull In fact gravitational waves have not been detected yet even if there are strong
indications that gravitational waves exist as predicted by general relativity
bull Such an indirect evidence comes from binary pulsars Two neutron stars orbiting each
other at least one of which is a pulsar The curvature is large so it is an interesting place
to study general relativistic effects (Incidentally their orbit may precess by 4 degree per
year)bull The spin axis of pulsars is not aligned with
their magnetic axis so at each rotation theyemit pulses of radio waves and we can
detect them from the Earth Pulsars behave
as very accurate clocks
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bull Actually such measurements are very precise and
confirm that the orbit is shrinking with time at a rate
in complete agreement with the emission of
gravitational waves predicted by general relativity
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radio
far-IR
mid-IR
near-IR
opticalx-ray
gamma-ray
GWs
It has happened over and over in the history of astronomy as a newldquowindowrdquo has been opened a ldquonewrdquo universe has been revealed
The same will happen with GW-astronomy
GSFCNASA
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Large-scale
Cryogenic
Gravitational-waveTelescope
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o Binary neutron stars
o Binary black-holes
o Deformed compact stars including Low Mass X-ray Binaries (LMXB)and pulsars
o Cosmological stochastic background
o Mixed binary systems
o Gravitational collapse (supernovae neutron stars)
o Extreme-Mass-Ratio Inspirals (EMRI)
o
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TAMA-Tokyo GEO-Hannover LIGO-Livingston LIGO-Hanford VIRGO-Cascina
and LCGT will measure LL ~ h lt 10-21 with SN~1
Numericalrelativity
h
Knowledge of thewaveforms cancompensate for the
very small SN(matched-filtering) and
so enhance detection and allow for source-characterization
possible
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ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
The fundamental equations standard numerical relativity aims at solving
are the Einstein equations
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General relativity states that our World is a 4D and curvedspacetime and the Einstein equations describe its dynamics
How to solve the Einstein equations numerically
Prominently there is no a priori concept of ldquoflowing of timerdquo (we arenot involved in thermodynamics here) time is just one of thedimensions and on the same level as space dimensions
There is a successful recipe though
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We have the illusion to live in 3D and it is easier to tell computers toperform simulations (time-)step by (time-)step
Also assign a normalizationsuch that
Define therefore
So given a manifold describing a spacetime with 4-metricwe want to foliate it via space-like three-dimensionalhypersurfaces We label such hypersurfaces with the
time coordinate t
(ldquothe direction of timerdquo)
(As mostly in numer ical relativity the signature is here -+++)
The function is called the rdquolapserdquo function and itis strictly positive for spacelike hypersurfaces
α
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so that
ii) and the spatial metric
Letrsquos also define
i) the unit normal vector to the hypersurface
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The spatial part is obtained by contracting with the spatialprojection operator defined as
By using we can decompose any 4D tensor into a
purely spatial part (hence in ) and a purely timelike part (henceorthogonal to and aligned with )
while the timelike part is obtained by contracting with the timelikeprojection operator
The two projectors are obviously orthogonal
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The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariantderivative
All the 4D tensors in the Einstein equations can be projectedstraightforwardly onto the 3D spatial slice
In particular the 3D connection coefficients
the 3D Riemann tensor
and the 3D contractions of the 3D Riemann tensor ie the 3DRicci tensor the 3D Ricci scalar and
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Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic
curvatureThe extrinsic curvature is defined in terms of the unit normal to as
where is the Lie derivative along
This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric
We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic
curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold
and this can be shown to be equivalent also to
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Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
8182019 Luca Bai Otti
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
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Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
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W f l i EOS
8182019 Luca Bai Otti
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
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T R3(GM )
part 3Qij
part t3
MR2
T 3
Mv2
T
LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
LGW = 15Gc5
part 3
Qij
part t3part 3
Qij
part t3 1048575
8182019 Luca Bai Otti
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LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
G4
c5
82times
10
minus74
cgs units
c5 27 times 10minus60 cgs units
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 20100
bull In fact gravitational waves have not been detected yet even if there are strong
indications that gravitational waves exist as predicted by general relativity
bull Such an indirect evidence comes from binary pulsars Two neutron stars orbiting each
other at least one of which is a pulsar The curvature is large so it is an interesting place
to study general relativistic effects (Incidentally their orbit may precess by 4 degree per
year)bull The spin axis of pulsars is not aligned with
their magnetic axis so at each rotation theyemit pulses of radio waves and we can
detect them from the Earth Pulsars behave
as very accurate clocks
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bull Actually such measurements are very precise and
confirm that the orbit is shrinking with time at a rate
in complete agreement with the emission of
gravitational waves predicted by general relativity
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radio
far-IR
mid-IR
near-IR
opticalx-ray
gamma-ray
GWs
It has happened over and over in the history of astronomy as a newldquowindowrdquo has been opened a ldquonewrdquo universe has been revealed
The same will happen with GW-astronomy
GSFCNASA
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Large-scale
Cryogenic
Gravitational-waveTelescope
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o Binary neutron stars
o Binary black-holes
o Deformed compact stars including Low Mass X-ray Binaries (LMXB)and pulsars
o Cosmological stochastic background
o Mixed binary systems
o Gravitational collapse (supernovae neutron stars)
o Extreme-Mass-Ratio Inspirals (EMRI)
o
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httpslidepdfcomreaderfullluca-bai-otti 27100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 28100
TAMA-Tokyo GEO-Hannover LIGO-Livingston LIGO-Hanford VIRGO-Cascina
and LCGT will measure LL ~ h lt 10-21 with SN~1
Numericalrelativity
h
Knowledge of thewaveforms cancompensate for the
very small SN(matched-filtering) and
so enhance detection and allow for source-characterization
possible
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httpslidepdfcomreaderfullluca-bai-otti 29100
ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
The fundamental equations standard numerical relativity aims at solving
are the Einstein equations
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General relativity states that our World is a 4D and curvedspacetime and the Einstein equations describe its dynamics
How to solve the Einstein equations numerically
Prominently there is no a priori concept of ldquoflowing of timerdquo (we arenot involved in thermodynamics here) time is just one of thedimensions and on the same level as space dimensions
There is a successful recipe though
8182019 Luca Bai Otti
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We have the illusion to live in 3D and it is easier to tell computers toperform simulations (time-)step by (time-)step
Also assign a normalizationsuch that
Define therefore
So given a manifold describing a spacetime with 4-metricwe want to foliate it via space-like three-dimensionalhypersurfaces We label such hypersurfaces with the
time coordinate t
(ldquothe direction of timerdquo)
(As mostly in numer ical relativity the signature is here -+++)
The function is called the rdquolapserdquo function and itis strictly positive for spacelike hypersurfaces
α
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so that
ii) and the spatial metric
Letrsquos also define
i) the unit normal vector to the hypersurface
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The spatial part is obtained by contracting with the spatialprojection operator defined as
By using we can decompose any 4D tensor into a
purely spatial part (hence in ) and a purely timelike part (henceorthogonal to and aligned with )
while the timelike part is obtained by contracting with the timelikeprojection operator
The two projectors are obviously orthogonal
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The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariantderivative
All the 4D tensors in the Einstein equations can be projectedstraightforwardly onto the 3D spatial slice
In particular the 3D connection coefficients
the 3D Riemann tensor
and the 3D contractions of the 3D Riemann tensor ie the 3DRicci tensor the 3D Ricci scalar and
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Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic
curvatureThe extrinsic curvature is defined in terms of the unit normal to as
where is the Lie derivative along
This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric
We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic
curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold
and this can be shown to be equivalent also to
8182019 Luca Bai Otti
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Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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httpslidepdfcomreaderfullluca-bai-otti 38100
Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
8182019 Luca Bai Otti
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
8182019 Luca Bai Otti
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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httpslidepdfcomreaderfullluca-bai-otti 48100
As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
8182019 Luca Bai Otti
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
8182019 Luca Bai Otti
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
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Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
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M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
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W f l i EOS
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 19100
LGW 1
5
G
c5
M 1048575v21114111
T
1
5
G4
c5
M
R
2
G4
c5
82times
10
minus74
cgs units
c5 27 times 10minus60 cgs units
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 20100
bull In fact gravitational waves have not been detected yet even if there are strong
indications that gravitational waves exist as predicted by general relativity
bull Such an indirect evidence comes from binary pulsars Two neutron stars orbiting each
other at least one of which is a pulsar The curvature is large so it is an interesting place
to study general relativistic effects (Incidentally their orbit may precess by 4 degree per
year)bull The spin axis of pulsars is not aligned with
their magnetic axis so at each rotation theyemit pulses of radio waves and we can
detect them from the Earth Pulsars behave
as very accurate clocks
8182019 Luca Bai Otti
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bull Actually such measurements are very precise and
confirm that the orbit is shrinking with time at a rate
in complete agreement with the emission of
gravitational waves predicted by general relativity
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radio
far-IR
mid-IR
near-IR
opticalx-ray
gamma-ray
GWs
It has happened over and over in the history of astronomy as a newldquowindowrdquo has been opened a ldquonewrdquo universe has been revealed
The same will happen with GW-astronomy
GSFCNASA
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httpslidepdfcomreaderfullluca-bai-otti 24100
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Large-scale
Cryogenic
Gravitational-waveTelescope
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o Binary neutron stars
o Binary black-holes
o Deformed compact stars including Low Mass X-ray Binaries (LMXB)and pulsars
o Cosmological stochastic background
o Mixed binary systems
o Gravitational collapse (supernovae neutron stars)
o Extreme-Mass-Ratio Inspirals (EMRI)
o
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TAMA-Tokyo GEO-Hannover LIGO-Livingston LIGO-Hanford VIRGO-Cascina
and LCGT will measure LL ~ h lt 10-21 with SN~1
Numericalrelativity
h
Knowledge of thewaveforms cancompensate for the
very small SN(matched-filtering) and
so enhance detection and allow for source-characterization
possible
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ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
The fundamental equations standard numerical relativity aims at solving
are the Einstein equations
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General relativity states that our World is a 4D and curvedspacetime and the Einstein equations describe its dynamics
How to solve the Einstein equations numerically
Prominently there is no a priori concept of ldquoflowing of timerdquo (we arenot involved in thermodynamics here) time is just one of thedimensions and on the same level as space dimensions
There is a successful recipe though
8182019 Luca Bai Otti
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We have the illusion to live in 3D and it is easier to tell computers toperform simulations (time-)step by (time-)step
Also assign a normalizationsuch that
Define therefore
So given a manifold describing a spacetime with 4-metricwe want to foliate it via space-like three-dimensionalhypersurfaces We label such hypersurfaces with the
time coordinate t
(ldquothe direction of timerdquo)
(As mostly in numer ical relativity the signature is here -+++)
The function is called the rdquolapserdquo function and itis strictly positive for spacelike hypersurfaces
α
8182019 Luca Bai Otti
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so that
ii) and the spatial metric
Letrsquos also define
i) the unit normal vector to the hypersurface
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The spatial part is obtained by contracting with the spatialprojection operator defined as
By using we can decompose any 4D tensor into a
purely spatial part (hence in ) and a purely timelike part (henceorthogonal to and aligned with )
while the timelike part is obtained by contracting with the timelikeprojection operator
The two projectors are obviously orthogonal
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The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariantderivative
All the 4D tensors in the Einstein equations can be projectedstraightforwardly onto the 3D spatial slice
In particular the 3D connection coefficients
the 3D Riemann tensor
and the 3D contractions of the 3D Riemann tensor ie the 3DRicci tensor the 3D Ricci scalar and
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Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic
curvatureThe extrinsic curvature is defined in terms of the unit normal to as
where is the Lie derivative along
This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric
We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic
curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold
and this can be shown to be equivalent also to
8182019 Luca Bai Otti
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Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
8182019 Luca Bai Otti
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
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Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 78100
T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
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8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
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W f l i EOS
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
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bull In fact gravitational waves have not been detected yet even if there are strong
indications that gravitational waves exist as predicted by general relativity
bull Such an indirect evidence comes from binary pulsars Two neutron stars orbiting each
other at least one of which is a pulsar The curvature is large so it is an interesting place
to study general relativistic effects (Incidentally their orbit may precess by 4 degree per
year)bull The spin axis of pulsars is not aligned with
their magnetic axis so at each rotation theyemit pulses of radio waves and we can
detect them from the Earth Pulsars behave
as very accurate clocks
8182019 Luca Bai Otti
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bull Actually such measurements are very precise and
confirm that the orbit is shrinking with time at a rate
in complete agreement with the emission of
gravitational waves predicted by general relativity
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radio
far-IR
mid-IR
near-IR
opticalx-ray
gamma-ray
GWs
It has happened over and over in the history of astronomy as a newldquowindowrdquo has been opened a ldquonewrdquo universe has been revealed
The same will happen with GW-astronomy
GSFCNASA
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Large-scale
Cryogenic
Gravitational-waveTelescope
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o Binary neutron stars
o Binary black-holes
o Deformed compact stars including Low Mass X-ray Binaries (LMXB)and pulsars
o Cosmological stochastic background
o Mixed binary systems
o Gravitational collapse (supernovae neutron stars)
o Extreme-Mass-Ratio Inspirals (EMRI)
o
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TAMA-Tokyo GEO-Hannover LIGO-Livingston LIGO-Hanford VIRGO-Cascina
and LCGT will measure LL ~ h lt 10-21 with SN~1
Numericalrelativity
h
Knowledge of thewaveforms cancompensate for the
very small SN(matched-filtering) and
so enhance detection and allow for source-characterization
possible
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ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
The fundamental equations standard numerical relativity aims at solving
are the Einstein equations
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General relativity states that our World is a 4D and curvedspacetime and the Einstein equations describe its dynamics
How to solve the Einstein equations numerically
Prominently there is no a priori concept of ldquoflowing of timerdquo (we arenot involved in thermodynamics here) time is just one of thedimensions and on the same level as space dimensions
There is a successful recipe though
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We have the illusion to live in 3D and it is easier to tell computers toperform simulations (time-)step by (time-)step
Also assign a normalizationsuch that
Define therefore
So given a manifold describing a spacetime with 4-metricwe want to foliate it via space-like three-dimensionalhypersurfaces We label such hypersurfaces with the
time coordinate t
(ldquothe direction of timerdquo)
(As mostly in numer ical relativity the signature is here -+++)
The function is called the rdquolapserdquo function and itis strictly positive for spacelike hypersurfaces
α
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so that
ii) and the spatial metric
Letrsquos also define
i) the unit normal vector to the hypersurface
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The spatial part is obtained by contracting with the spatialprojection operator defined as
By using we can decompose any 4D tensor into a
purely spatial part (hence in ) and a purely timelike part (henceorthogonal to and aligned with )
while the timelike part is obtained by contracting with the timelikeprojection operator
The two projectors are obviously orthogonal
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The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariantderivative
All the 4D tensors in the Einstein equations can be projectedstraightforwardly onto the 3D spatial slice
In particular the 3D connection coefficients
the 3D Riemann tensor
and the 3D contractions of the 3D Riemann tensor ie the 3DRicci tensor the 3D Ricci scalar and
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Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic
curvatureThe extrinsic curvature is defined in terms of the unit normal to as
where is the Lie derivative along
This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric
We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic
curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold
and this can be shown to be equivalent also to
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Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 67100
conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 68100
Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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httpslidepdfcomreaderfullluca-bai-otti 71100
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
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W f l i EOS
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
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bull Actually such measurements are very precise and
confirm that the orbit is shrinking with time at a rate
in complete agreement with the emission of
gravitational waves predicted by general relativity
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radio
far-IR
mid-IR
near-IR
opticalx-ray
gamma-ray
GWs
It has happened over and over in the history of astronomy as a newldquowindowrdquo has been opened a ldquonewrdquo universe has been revealed
The same will happen with GW-astronomy
GSFCNASA
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Large-scale
Cryogenic
Gravitational-waveTelescope
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o Binary neutron stars
o Binary black-holes
o Deformed compact stars including Low Mass X-ray Binaries (LMXB)and pulsars
o Cosmological stochastic background
o Mixed binary systems
o Gravitational collapse (supernovae neutron stars)
o Extreme-Mass-Ratio Inspirals (EMRI)
o
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TAMA-Tokyo GEO-Hannover LIGO-Livingston LIGO-Hanford VIRGO-Cascina
and LCGT will measure LL ~ h lt 10-21 with SN~1
Numericalrelativity
h
Knowledge of thewaveforms cancompensate for the
very small SN(matched-filtering) and
so enhance detection and allow for source-characterization
possible
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ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
The fundamental equations standard numerical relativity aims at solving
are the Einstein equations
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General relativity states that our World is a 4D and curvedspacetime and the Einstein equations describe its dynamics
How to solve the Einstein equations numerically
Prominently there is no a priori concept of ldquoflowing of timerdquo (we arenot involved in thermodynamics here) time is just one of thedimensions and on the same level as space dimensions
There is a successful recipe though
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We have the illusion to live in 3D and it is easier to tell computers toperform simulations (time-)step by (time-)step
Also assign a normalizationsuch that
Define therefore
So given a manifold describing a spacetime with 4-metricwe want to foliate it via space-like three-dimensionalhypersurfaces We label such hypersurfaces with the
time coordinate t
(ldquothe direction of timerdquo)
(As mostly in numer ical relativity the signature is here -+++)
The function is called the rdquolapserdquo function and itis strictly positive for spacelike hypersurfaces
α
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so that
ii) and the spatial metric
Letrsquos also define
i) the unit normal vector to the hypersurface
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The spatial part is obtained by contracting with the spatialprojection operator defined as
By using we can decompose any 4D tensor into a
purely spatial part (hence in ) and a purely timelike part (henceorthogonal to and aligned with )
while the timelike part is obtained by contracting with the timelikeprojection operator
The two projectors are obviously orthogonal
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The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariantderivative
All the 4D tensors in the Einstein equations can be projectedstraightforwardly onto the 3D spatial slice
In particular the 3D connection coefficients
the 3D Riemann tensor
and the 3D contractions of the 3D Riemann tensor ie the 3DRicci tensor the 3D Ricci scalar and
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Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic
curvatureThe extrinsic curvature is defined in terms of the unit normal to as
where is the Lie derivative along
This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric
We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic
curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold
and this can be shown to be equivalent also to
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Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
8182019 Luca Bai Otti
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 59100
where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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httpslidepdfcomreaderfullluca-bai-otti 60100
In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
8182019 Luca Bai Otti
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
8182019 Luca Bai Otti
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 66100
where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 67100
conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 68100
Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 72100
Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
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W f l i EOS
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
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radio
far-IR
mid-IR
near-IR
opticalx-ray
gamma-ray
GWs
It has happened over and over in the history of astronomy as a newldquowindowrdquo has been opened a ldquonewrdquo universe has been revealed
The same will happen with GW-astronomy
GSFCNASA
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Large-scale
Cryogenic
Gravitational-waveTelescope
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o Binary neutron stars
o Binary black-holes
o Deformed compact stars including Low Mass X-ray Binaries (LMXB)and pulsars
o Cosmological stochastic background
o Mixed binary systems
o Gravitational collapse (supernovae neutron stars)
o Extreme-Mass-Ratio Inspirals (EMRI)
o
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TAMA-Tokyo GEO-Hannover LIGO-Livingston LIGO-Hanford VIRGO-Cascina
and LCGT will measure LL ~ h lt 10-21 with SN~1
Numericalrelativity
h
Knowledge of thewaveforms cancompensate for the
very small SN(matched-filtering) and
so enhance detection and allow for source-characterization
possible
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ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
The fundamental equations standard numerical relativity aims at solving
are the Einstein equations
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General relativity states that our World is a 4D and curvedspacetime and the Einstein equations describe its dynamics
How to solve the Einstein equations numerically
Prominently there is no a priori concept of ldquoflowing of timerdquo (we arenot involved in thermodynamics here) time is just one of thedimensions and on the same level as space dimensions
There is a successful recipe though
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We have the illusion to live in 3D and it is easier to tell computers toperform simulations (time-)step by (time-)step
Also assign a normalizationsuch that
Define therefore
So given a manifold describing a spacetime with 4-metricwe want to foliate it via space-like three-dimensionalhypersurfaces We label such hypersurfaces with the
time coordinate t
(ldquothe direction of timerdquo)
(As mostly in numer ical relativity the signature is here -+++)
The function is called the rdquolapserdquo function and itis strictly positive for spacelike hypersurfaces
α
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so that
ii) and the spatial metric
Letrsquos also define
i) the unit normal vector to the hypersurface
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The spatial part is obtained by contracting with the spatialprojection operator defined as
By using we can decompose any 4D tensor into a
purely spatial part (hence in ) and a purely timelike part (henceorthogonal to and aligned with )
while the timelike part is obtained by contracting with the timelikeprojection operator
The two projectors are obviously orthogonal
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The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariantderivative
All the 4D tensors in the Einstein equations can be projectedstraightforwardly onto the 3D spatial slice
In particular the 3D connection coefficients
the 3D Riemann tensor
and the 3D contractions of the 3D Riemann tensor ie the 3DRicci tensor the 3D Ricci scalar and
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Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic
curvatureThe extrinsic curvature is defined in terms of the unit normal to as
where is the Lie derivative along
This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric
We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic
curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold
and this can be shown to be equivalent also to
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Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
8182019 Luca Bai Otti
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
8182019 Luca Bai Otti
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
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Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
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W f l i EOS
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
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radio
far-IR
mid-IR
near-IR
opticalx-ray
gamma-ray
GWs
It has happened over and over in the history of astronomy as a newldquowindowrdquo has been opened a ldquonewrdquo universe has been revealed
The same will happen with GW-astronomy
GSFCNASA
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Large-scale
Cryogenic
Gravitational-waveTelescope
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o Binary neutron stars
o Binary black-holes
o Deformed compact stars including Low Mass X-ray Binaries (LMXB)and pulsars
o Cosmological stochastic background
o Mixed binary systems
o Gravitational collapse (supernovae neutron stars)
o Extreme-Mass-Ratio Inspirals (EMRI)
o
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TAMA-Tokyo GEO-Hannover LIGO-Livingston LIGO-Hanford VIRGO-Cascina
and LCGT will measure LL ~ h lt 10-21 with SN~1
Numericalrelativity
h
Knowledge of thewaveforms cancompensate for the
very small SN(matched-filtering) and
so enhance detection and allow for source-characterization
possible
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ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
The fundamental equations standard numerical relativity aims at solving
are the Einstein equations
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General relativity states that our World is a 4D and curvedspacetime and the Einstein equations describe its dynamics
How to solve the Einstein equations numerically
Prominently there is no a priori concept of ldquoflowing of timerdquo (we arenot involved in thermodynamics here) time is just one of thedimensions and on the same level as space dimensions
There is a successful recipe though
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We have the illusion to live in 3D and it is easier to tell computers toperform simulations (time-)step by (time-)step
Also assign a normalizationsuch that
Define therefore
So given a manifold describing a spacetime with 4-metricwe want to foliate it via space-like three-dimensionalhypersurfaces We label such hypersurfaces with the
time coordinate t
(ldquothe direction of timerdquo)
(As mostly in numer ical relativity the signature is here -+++)
The function is called the rdquolapserdquo function and itis strictly positive for spacelike hypersurfaces
α
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so that
ii) and the spatial metric
Letrsquos also define
i) the unit normal vector to the hypersurface
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The spatial part is obtained by contracting with the spatialprojection operator defined as
By using we can decompose any 4D tensor into a
purely spatial part (hence in ) and a purely timelike part (henceorthogonal to and aligned with )
while the timelike part is obtained by contracting with the timelikeprojection operator
The two projectors are obviously orthogonal
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The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariantderivative
All the 4D tensors in the Einstein equations can be projectedstraightforwardly onto the 3D spatial slice
In particular the 3D connection coefficients
the 3D Riemann tensor
and the 3D contractions of the 3D Riemann tensor ie the 3DRicci tensor the 3D Ricci scalar and
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Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic
curvatureThe extrinsic curvature is defined in terms of the unit normal to as
where is the Lie derivative along
This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric
We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic
curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold
and this can be shown to be equivalent also to
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Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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httpslidepdfcomreaderfullluca-bai-otti 38100
Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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httpslidepdfcomreaderfullluca-bai-otti 41100
Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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httpslidepdfcomreaderfullluca-bai-otti 46100
We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
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Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
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M d i
8182019 Luca Bai Otti
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
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W f l i EOS
8182019 Luca Bai Otti
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
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Large-scale
Cryogenic
Gravitational-waveTelescope
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o Binary neutron stars
o Binary black-holes
o Deformed compact stars including Low Mass X-ray Binaries (LMXB)and pulsars
o Cosmological stochastic background
o Mixed binary systems
o Gravitational collapse (supernovae neutron stars)
o Extreme-Mass-Ratio Inspirals (EMRI)
o
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TAMA-Tokyo GEO-Hannover LIGO-Livingston LIGO-Hanford VIRGO-Cascina
and LCGT will measure LL ~ h lt 10-21 with SN~1
Numericalrelativity
h
Knowledge of thewaveforms cancompensate for the
very small SN(matched-filtering) and
so enhance detection and allow for source-characterization
possible
8182019 Luca Bai Otti
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ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
The fundamental equations standard numerical relativity aims at solving
are the Einstein equations
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General relativity states that our World is a 4D and curvedspacetime and the Einstein equations describe its dynamics
How to solve the Einstein equations numerically
Prominently there is no a priori concept of ldquoflowing of timerdquo (we arenot involved in thermodynamics here) time is just one of thedimensions and on the same level as space dimensions
There is a successful recipe though
8182019 Luca Bai Otti
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We have the illusion to live in 3D and it is easier to tell computers toperform simulations (time-)step by (time-)step
Also assign a normalizationsuch that
Define therefore
So given a manifold describing a spacetime with 4-metricwe want to foliate it via space-like three-dimensionalhypersurfaces We label such hypersurfaces with the
time coordinate t
(ldquothe direction of timerdquo)
(As mostly in numer ical relativity the signature is here -+++)
The function is called the rdquolapserdquo function and itis strictly positive for spacelike hypersurfaces
α
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so that
ii) and the spatial metric
Letrsquos also define
i) the unit normal vector to the hypersurface
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The spatial part is obtained by contracting with the spatialprojection operator defined as
By using we can decompose any 4D tensor into a
purely spatial part (hence in ) and a purely timelike part (henceorthogonal to and aligned with )
while the timelike part is obtained by contracting with the timelikeprojection operator
The two projectors are obviously orthogonal
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The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariantderivative
All the 4D tensors in the Einstein equations can be projectedstraightforwardly onto the 3D spatial slice
In particular the 3D connection coefficients
the 3D Riemann tensor
and the 3D contractions of the 3D Riemann tensor ie the 3DRicci tensor the 3D Ricci scalar and
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Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic
curvatureThe extrinsic curvature is defined in terms of the unit normal to as
where is the Lie derivative along
This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric
We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic
curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold
and this can be shown to be equivalent also to
8182019 Luca Bai Otti
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Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
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Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
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W f l i EOS
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
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httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
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Large-scale
Cryogenic
Gravitational-waveTelescope
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o Binary neutron stars
o Binary black-holes
o Deformed compact stars including Low Mass X-ray Binaries (LMXB)and pulsars
o Cosmological stochastic background
o Mixed binary systems
o Gravitational collapse (supernovae neutron stars)
o Extreme-Mass-Ratio Inspirals (EMRI)
o
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httpslidepdfcomreaderfullluca-bai-otti 27100
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httpslidepdfcomreaderfullluca-bai-otti 28100
TAMA-Tokyo GEO-Hannover LIGO-Livingston LIGO-Hanford VIRGO-Cascina
and LCGT will measure LL ~ h lt 10-21 with SN~1
Numericalrelativity
h
Knowledge of thewaveforms cancompensate for the
very small SN(matched-filtering) and
so enhance detection and allow for source-characterization
possible
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httpslidepdfcomreaderfullluca-bai-otti 29100
ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
The fundamental equations standard numerical relativity aims at solving
are the Einstein equations
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General relativity states that our World is a 4D and curvedspacetime and the Einstein equations describe its dynamics
How to solve the Einstein equations numerically
Prominently there is no a priori concept of ldquoflowing of timerdquo (we arenot involved in thermodynamics here) time is just one of thedimensions and on the same level as space dimensions
There is a successful recipe though
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 31100
We have the illusion to live in 3D and it is easier to tell computers toperform simulations (time-)step by (time-)step
Also assign a normalizationsuch that
Define therefore
So given a manifold describing a spacetime with 4-metricwe want to foliate it via space-like three-dimensionalhypersurfaces We label such hypersurfaces with the
time coordinate t
(ldquothe direction of timerdquo)
(As mostly in numer ical relativity the signature is here -+++)
The function is called the rdquolapserdquo function and itis strictly positive for spacelike hypersurfaces
α
8182019 Luca Bai Otti
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so that
ii) and the spatial metric
Letrsquos also define
i) the unit normal vector to the hypersurface
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The spatial part is obtained by contracting with the spatialprojection operator defined as
By using we can decompose any 4D tensor into a
purely spatial part (hence in ) and a purely timelike part (henceorthogonal to and aligned with )
while the timelike part is obtained by contracting with the timelikeprojection operator
The two projectors are obviously orthogonal
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The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariantderivative
All the 4D tensors in the Einstein equations can be projectedstraightforwardly onto the 3D spatial slice
In particular the 3D connection coefficients
the 3D Riemann tensor
and the 3D contractions of the 3D Riemann tensor ie the 3DRicci tensor the 3D Ricci scalar and
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httpslidepdfcomreaderfullluca-bai-otti 35100
Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic
curvatureThe extrinsic curvature is defined in terms of the unit normal to as
where is the Lie derivative along
This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric
We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic
curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold
and this can be shown to be equivalent also to
8182019 Luca Bai Otti
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Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
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Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
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W f l i EOS
8182019 Luca Bai Otti
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
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httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 26100
o Binary neutron stars
o Binary black-holes
o Deformed compact stars including Low Mass X-ray Binaries (LMXB)and pulsars
o Cosmological stochastic background
o Mixed binary systems
o Gravitational collapse (supernovae neutron stars)
o Extreme-Mass-Ratio Inspirals (EMRI)
o
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httpslidepdfcomreaderfullluca-bai-otti 27100
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httpslidepdfcomreaderfullluca-bai-otti 28100
TAMA-Tokyo GEO-Hannover LIGO-Livingston LIGO-Hanford VIRGO-Cascina
and LCGT will measure LL ~ h lt 10-21 with SN~1
Numericalrelativity
h
Knowledge of thewaveforms cancompensate for the
very small SN(matched-filtering) and
so enhance detection and allow for source-characterization
possible
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ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
The fundamental equations standard numerical relativity aims at solving
are the Einstein equations
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General relativity states that our World is a 4D and curvedspacetime and the Einstein equations describe its dynamics
How to solve the Einstein equations numerically
Prominently there is no a priori concept of ldquoflowing of timerdquo (we arenot involved in thermodynamics here) time is just one of thedimensions and on the same level as space dimensions
There is a successful recipe though
8182019 Luca Bai Otti
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We have the illusion to live in 3D and it is easier to tell computers toperform simulations (time-)step by (time-)step
Also assign a normalizationsuch that
Define therefore
So given a manifold describing a spacetime with 4-metricwe want to foliate it via space-like three-dimensionalhypersurfaces We label such hypersurfaces with the
time coordinate t
(ldquothe direction of timerdquo)
(As mostly in numer ical relativity the signature is here -+++)
The function is called the rdquolapserdquo function and itis strictly positive for spacelike hypersurfaces
α
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so that
ii) and the spatial metric
Letrsquos also define
i) the unit normal vector to the hypersurface
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The spatial part is obtained by contracting with the spatialprojection operator defined as
By using we can decompose any 4D tensor into a
purely spatial part (hence in ) and a purely timelike part (henceorthogonal to and aligned with )
while the timelike part is obtained by contracting with the timelikeprojection operator
The two projectors are obviously orthogonal
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The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariantderivative
All the 4D tensors in the Einstein equations can be projectedstraightforwardly onto the 3D spatial slice
In particular the 3D connection coefficients
the 3D Riemann tensor
and the 3D contractions of the 3D Riemann tensor ie the 3DRicci tensor the 3D Ricci scalar and
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Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic
curvatureThe extrinsic curvature is defined in terms of the unit normal to as
where is the Lie derivative along
This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric
We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic
curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold
and this can be shown to be equivalent also to
8182019 Luca Bai Otti
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Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
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Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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8182019 Luca Bai Otti
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
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W f l i EOS
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
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TAMA-Tokyo GEO-Hannover LIGO-Livingston LIGO-Hanford VIRGO-Cascina
and LCGT will measure LL ~ h lt 10-21 with SN~1
Numericalrelativity
h
Knowledge of thewaveforms cancompensate for the
very small SN(matched-filtering) and
so enhance detection and allow for source-characterization
possible
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ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
The fundamental equations standard numerical relativity aims at solving
are the Einstein equations
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General relativity states that our World is a 4D and curvedspacetime and the Einstein equations describe its dynamics
How to solve the Einstein equations numerically
Prominently there is no a priori concept of ldquoflowing of timerdquo (we arenot involved in thermodynamics here) time is just one of thedimensions and on the same level as space dimensions
There is a successful recipe though
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We have the illusion to live in 3D and it is easier to tell computers toperform simulations (time-)step by (time-)step
Also assign a normalizationsuch that
Define therefore
So given a manifold describing a spacetime with 4-metricwe want to foliate it via space-like three-dimensionalhypersurfaces We label such hypersurfaces with the
time coordinate t
(ldquothe direction of timerdquo)
(As mostly in numer ical relativity the signature is here -+++)
The function is called the rdquolapserdquo function and itis strictly positive for spacelike hypersurfaces
α
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so that
ii) and the spatial metric
Letrsquos also define
i) the unit normal vector to the hypersurface
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The spatial part is obtained by contracting with the spatialprojection operator defined as
By using we can decompose any 4D tensor into a
purely spatial part (hence in ) and a purely timelike part (henceorthogonal to and aligned with )
while the timelike part is obtained by contracting with the timelikeprojection operator
The two projectors are obviously orthogonal
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The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariantderivative
All the 4D tensors in the Einstein equations can be projectedstraightforwardly onto the 3D spatial slice
In particular the 3D connection coefficients
the 3D Riemann tensor
and the 3D contractions of the 3D Riemann tensor ie the 3DRicci tensor the 3D Ricci scalar and
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Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic
curvatureThe extrinsic curvature is defined in terms of the unit normal to as
where is the Lie derivative along
This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric
We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic
curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold
and this can be shown to be equivalent also to
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Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
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Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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httpslidepdfcomreaderfullluca-bai-otti 71100
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
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W f l i EOS
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
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TAMA-Tokyo GEO-Hannover LIGO-Livingston LIGO-Hanford VIRGO-Cascina
and LCGT will measure LL ~ h lt 10-21 with SN~1
Numericalrelativity
h
Knowledge of thewaveforms cancompensate for the
very small SN(matched-filtering) and
so enhance detection and allow for source-characterization
possible
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ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
The fundamental equations standard numerical relativity aims at solving
are the Einstein equations
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General relativity states that our World is a 4D and curvedspacetime and the Einstein equations describe its dynamics
How to solve the Einstein equations numerically
Prominently there is no a priori concept of ldquoflowing of timerdquo (we arenot involved in thermodynamics here) time is just one of thedimensions and on the same level as space dimensions
There is a successful recipe though
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We have the illusion to live in 3D and it is easier to tell computers toperform simulations (time-)step by (time-)step
Also assign a normalizationsuch that
Define therefore
So given a manifold describing a spacetime with 4-metricwe want to foliate it via space-like three-dimensionalhypersurfaces We label such hypersurfaces with the
time coordinate t
(ldquothe direction of timerdquo)
(As mostly in numer ical relativity the signature is here -+++)
The function is called the rdquolapserdquo function and itis strictly positive for spacelike hypersurfaces
α
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so that
ii) and the spatial metric
Letrsquos also define
i) the unit normal vector to the hypersurface
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The spatial part is obtained by contracting with the spatialprojection operator defined as
By using we can decompose any 4D tensor into a
purely spatial part (hence in ) and a purely timelike part (henceorthogonal to and aligned with )
while the timelike part is obtained by contracting with the timelikeprojection operator
The two projectors are obviously orthogonal
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The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariantderivative
All the 4D tensors in the Einstein equations can be projectedstraightforwardly onto the 3D spatial slice
In particular the 3D connection coefficients
the 3D Riemann tensor
and the 3D contractions of the 3D Riemann tensor ie the 3DRicci tensor the 3D Ricci scalar and
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Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic
curvatureThe extrinsic curvature is defined in terms of the unit normal to as
where is the Lie derivative along
This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric
We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic
curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold
and this can be shown to be equivalent also to
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Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
8182019 Luca Bai Otti
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
8182019 Luca Bai Otti
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
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Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 72100
Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
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W f l i EOS
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
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ie six second-order-in-time second-order-in-space coupled highly-
nonlinear quasi-hyperbolic partial differential equations (PDEs)
four second-order-in-space coupled highly-nonlinear elliptic PDEs
Matter and other
fieldsCurvature scalar
Metric(measure of spacetime distances)
Einstein
tensor
(spacetime)
Rest-mass density Internal energy density Pressure
4-velocity
Ricci tensor
The fundamental equations standard numerical relativity aims at solving
are the Einstein equations
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General relativity states that our World is a 4D and curvedspacetime and the Einstein equations describe its dynamics
How to solve the Einstein equations numerically
Prominently there is no a priori concept of ldquoflowing of timerdquo (we arenot involved in thermodynamics here) time is just one of thedimensions and on the same level as space dimensions
There is a successful recipe though
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We have the illusion to live in 3D and it is easier to tell computers toperform simulations (time-)step by (time-)step
Also assign a normalizationsuch that
Define therefore
So given a manifold describing a spacetime with 4-metricwe want to foliate it via space-like three-dimensionalhypersurfaces We label such hypersurfaces with the
time coordinate t
(ldquothe direction of timerdquo)
(As mostly in numer ical relativity the signature is here -+++)
The function is called the rdquolapserdquo function and itis strictly positive for spacelike hypersurfaces
α
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so that
ii) and the spatial metric
Letrsquos also define
i) the unit normal vector to the hypersurface
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The spatial part is obtained by contracting with the spatialprojection operator defined as
By using we can decompose any 4D tensor into a
purely spatial part (hence in ) and a purely timelike part (henceorthogonal to and aligned with )
while the timelike part is obtained by contracting with the timelikeprojection operator
The two projectors are obviously orthogonal
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The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariantderivative
All the 4D tensors in the Einstein equations can be projectedstraightforwardly onto the 3D spatial slice
In particular the 3D connection coefficients
the 3D Riemann tensor
and the 3D contractions of the 3D Riemann tensor ie the 3DRicci tensor the 3D Ricci scalar and
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Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic
curvatureThe extrinsic curvature is defined in terms of the unit normal to as
where is the Lie derivative along
This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric
We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic
curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold
and this can be shown to be equivalent also to
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Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
8182019 Luca Bai Otti
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 59100
where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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httpslidepdfcomreaderfullluca-bai-otti 60100
In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
8182019 Luca Bai Otti
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
8182019 Luca Bai Otti
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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httpslidepdfcomreaderfullluca-bai-otti 63100
Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
8182019 Luca Bai Otti
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 66100
where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 68100
Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
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W f l i EOS
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
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General relativity states that our World is a 4D and curvedspacetime and the Einstein equations describe its dynamics
How to solve the Einstein equations numerically
Prominently there is no a priori concept of ldquoflowing of timerdquo (we arenot involved in thermodynamics here) time is just one of thedimensions and on the same level as space dimensions
There is a successful recipe though
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We have the illusion to live in 3D and it is easier to tell computers toperform simulations (time-)step by (time-)step
Also assign a normalizationsuch that
Define therefore
So given a manifold describing a spacetime with 4-metricwe want to foliate it via space-like three-dimensionalhypersurfaces We label such hypersurfaces with the
time coordinate t
(ldquothe direction of timerdquo)
(As mostly in numer ical relativity the signature is here -+++)
The function is called the rdquolapserdquo function and itis strictly positive for spacelike hypersurfaces
α
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so that
ii) and the spatial metric
Letrsquos also define
i) the unit normal vector to the hypersurface
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The spatial part is obtained by contracting with the spatialprojection operator defined as
By using we can decompose any 4D tensor into a
purely spatial part (hence in ) and a purely timelike part (henceorthogonal to and aligned with )
while the timelike part is obtained by contracting with the timelikeprojection operator
The two projectors are obviously orthogonal
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The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariantderivative
All the 4D tensors in the Einstein equations can be projectedstraightforwardly onto the 3D spatial slice
In particular the 3D connection coefficients
the 3D Riemann tensor
and the 3D contractions of the 3D Riemann tensor ie the 3DRicci tensor the 3D Ricci scalar and
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Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic
curvatureThe extrinsic curvature is defined in terms of the unit normal to as
where is the Lie derivative along
This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric
We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic
curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold
and this can be shown to be equivalent also to
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Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 57100
iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
8182019 Luca Bai Otti
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
8182019 Luca Bai Otti
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
8182019 Luca Bai Otti
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
8182019 Luca Bai Otti
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
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Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
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W f l i EOS
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
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We have the illusion to live in 3D and it is easier to tell computers toperform simulations (time-)step by (time-)step
Also assign a normalizationsuch that
Define therefore
So given a manifold describing a spacetime with 4-metricwe want to foliate it via space-like three-dimensionalhypersurfaces We label such hypersurfaces with the
time coordinate t
(ldquothe direction of timerdquo)
(As mostly in numer ical relativity the signature is here -+++)
The function is called the rdquolapserdquo function and itis strictly positive for spacelike hypersurfaces
α
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so that
ii) and the spatial metric
Letrsquos also define
i) the unit normal vector to the hypersurface
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The spatial part is obtained by contracting with the spatialprojection operator defined as
By using we can decompose any 4D tensor into a
purely spatial part (hence in ) and a purely timelike part (henceorthogonal to and aligned with )
while the timelike part is obtained by contracting with the timelikeprojection operator
The two projectors are obviously orthogonal
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The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariantderivative
All the 4D tensors in the Einstein equations can be projectedstraightforwardly onto the 3D spatial slice
In particular the 3D connection coefficients
the 3D Riemann tensor
and the 3D contractions of the 3D Riemann tensor ie the 3DRicci tensor the 3D Ricci scalar and
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Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic
curvatureThe extrinsic curvature is defined in terms of the unit normal to as
where is the Lie derivative along
This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric
We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic
curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold
and this can be shown to be equivalent also to
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Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
8182019 Luca Bai Otti
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 51100
We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
8182019 Luca Bai Otti
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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httpslidepdfcomreaderfullluca-bai-otti 57100
iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
8182019 Luca Bai Otti
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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httpslidepdfcomreaderfullluca-bai-otti 60100
In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
8182019 Luca Bai Otti
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
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Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
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W f l i EOS
8182019 Luca Bai Otti
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
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so that
ii) and the spatial metric
Letrsquos also define
i) the unit normal vector to the hypersurface
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The spatial part is obtained by contracting with the spatialprojection operator defined as
By using we can decompose any 4D tensor into a
purely spatial part (hence in ) and a purely timelike part (henceorthogonal to and aligned with )
while the timelike part is obtained by contracting with the timelikeprojection operator
The two projectors are obviously orthogonal
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The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariantderivative
All the 4D tensors in the Einstein equations can be projectedstraightforwardly onto the 3D spatial slice
In particular the 3D connection coefficients
the 3D Riemann tensor
and the 3D contractions of the 3D Riemann tensor ie the 3DRicci tensor the 3D Ricci scalar and
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Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic
curvatureThe extrinsic curvature is defined in terms of the unit normal to as
where is the Lie derivative along
This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric
We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic
curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold
and this can be shown to be equivalent also to
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Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
8182019 Luca Bai Otti
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
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Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
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W f l i EOS
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
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The spatial part is obtained by contracting with the spatialprojection operator defined as
By using we can decompose any 4D tensor into a
purely spatial part (hence in ) and a purely timelike part (henceorthogonal to and aligned with )
while the timelike part is obtained by contracting with the timelikeprojection operator
The two projectors are obviously orthogonal
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The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariantderivative
All the 4D tensors in the Einstein equations can be projectedstraightforwardly onto the 3D spatial slice
In particular the 3D connection coefficients
the 3D Riemann tensor
and the 3D contractions of the 3D Riemann tensor ie the 3DRicci tensor the 3D Ricci scalar and
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Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic
curvatureThe extrinsic curvature is defined in terms of the unit normal to as
where is the Lie derivative along
This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric
We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic
curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold
and this can be shown to be equivalent also to
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Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
8182019 Luca Bai Otti
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
8182019 Luca Bai Otti
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
8182019 Luca Bai Otti
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
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Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
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W f l i EOS
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
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The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariantderivative
All the 4D tensors in the Einstein equations can be projectedstraightforwardly onto the 3D spatial slice
In particular the 3D connection coefficients
the 3D Riemann tensor
and the 3D contractions of the 3D Riemann tensor ie the 3DRicci tensor the 3D Ricci scalar and
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Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic
curvatureThe extrinsic curvature is defined in terms of the unit normal to as
where is the Lie derivative along
This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric
We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic
curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold
and this can be shown to be equivalent also to
8182019 Luca Bai Otti
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Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
8182019 Luca Bai Otti
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
8182019 Luca Bai Otti
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
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Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
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W f l i EOS
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
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Another way of saying this is that the information present in(the 4D Riemann tensor) and ldquomissingrdquo in (the 3D version)can be found in another spatial tensor precisely the extrinsic
curvatureThe extrinsic curvature is defined in terms of the unit normal to as
where is the Lie derivative along
This also expresses that the extrinsic curvature can be seen as therate of change of the spatial metric
We have restricted ourselves to 3D hypersurfaces have we lostsome information about the 4D manifold (and so of full GR) Notreally such information is contained in a quantity called extrinsic
curvature which describes how the 3D hypersurface is embedded(ldquobentrdquo) in the 4D manifold
and this can be shown to be equivalent also to
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Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
8182019 Luca Bai Otti
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
8182019 Luca Bai Otti
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
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Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
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W f l i EOS
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
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Recall that the Lie derivative can be thought of as a geometricalgeneralization of a directional derivative It evaluates the change of a
tensor field along the flow of a vector fieldFor a scalar function this is given by
For a vector field this is given by the commutator
For a 1-form this is given by
As a result for a generic tensor of rank this is given by
= X micropart microV
ν
minus V micropart microX
ν
= X micropart microων + ωmicropart microX
ν
= X α
part αT micro
ν minus T
α
ν part αX
micro+ T
micro
αpart ν X
α
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
8182019 Luca Bai Otti
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
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Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
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W f l i EOS
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
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Consider a vector at one position and then parallel- transport it to a new location
The difference in the two vectors is proportional to theextrinsic curvature and this can either be positive or negative
P P + δ P
The extrinsic curvature measures the gradients of the normalvectors and since these are normalized they can only differ indirection Thus the extrinsic curvature provides information on howmuch the normal direction changes from point to point and so onhow the hypersurface is deformed
parallel
transport
Hence it measures how the 3Dhypersurface is ldquobentrdquo with
respect to the 4D spacetime
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Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
8182019 Luca Bai Otti
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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httpslidepdfcomreaderfullluca-bai-otti 51100
We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
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Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
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W f l i EOS
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
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Next we need to decompose the Einstein equations in the spatial
and timelike parts
To this purpose it is useful to derive a few identities
Gauss equations decompose the 4D Riemann tensorprojecting all indices
Codazzi equations take 3 spatial projections and a timelike one
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
8182019 Luca Bai Otti
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
8182019 Luca Bai Otti
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 56100
Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 57100
iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
8182019 Luca Bai Otti
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
8182019 Luca Bai Otti
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
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Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
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W f l i EOS
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
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Ricci equations take 2 spatial projections and 2 timelike ones
Another important identity which will be used in the following is
and which holds for any spatial vector
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
8182019 Luca Bai Otti
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 55100
Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 56100
Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 57100
iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
8182019 Luca Bai Otti
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
8182019 Luca Bai Otti
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 60100
In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
8182019 Luca Bai Otti
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
8182019 Luca Bai Otti
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
8182019 Luca Bai Otti
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
8182019 Luca Bai Otti
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
8182019 Luca Bai Otti
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 68100
Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
8182019 Luca Bai Otti
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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8182019 Luca Bai Otti
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
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M d i
8182019 Luca Bai Otti
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
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W f l i EOS
8182019 Luca Bai Otti
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
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Letrsquos define the double timelike projection of the stress-energy tensor as
We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations)
Similarly the momentum density (ie the mass current) will be givenby the mixed time and spatial projection
And similarly for the space-space projection
jmicro = minus
γ
α
micron
β
T αβ
8182019 Luca Bai Otti
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
8182019 Luca Bai Otti
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
8182019 Luca Bai Otti
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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8182019 Luca Bai Otti
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
8182019 Luca Bai Otti
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
8182019 Luca Bai Otti
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
8182019 Luca Bai Otti
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
8182019 Luca Bai Otti
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
8182019 Luca Bai Otti
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
8182019 Luca Bai Otti
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
8182019 Luca Bai Otti
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
8182019 Luca Bai Otti
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
8182019 Luca Bai Otti
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
8182019 Luca Bai Otti
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
8182019 Luca Bai Otti
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
8182019 Luca Bai Otti
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
8182019 Luca Bai Otti
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
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Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
8182019 Luca Bai Otti
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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8182019 Luca Bai Otti
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
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W f l i EOS
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
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Now we can decompose the Einstein equations in the 3+1splitting
We will get two sets of equations
1) the ldquoconstraintrdquo equations which are fully defined on eachspatial hypersurfaces (and do not involve time derivatives)2) the ldquoevolutionrdquo equations which instead relate quantities(the spatial metric and the extrinsic curvature) between twoadjacent hypersurfaces
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
8182019 Luca Bai Otti
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
8182019 Luca Bai Otti
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 68100
Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 73100
In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
8182019 Luca Bai Otti
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
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8182019 Luca Bai Otti
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
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M d i
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
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W f l i EOS
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
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We first time-project twice the left-hand-side of the Einstein
equations to obtain
Doing the same for the right-hand-side using the Gaussequations contracted twice with the spatial metric and the
definition of the energy density we finally reach the form of theequation which is called Hamiltonian constraint equation
Note that this is a single elliptic equation (not containing timederivatives) which should be satisfied everywhere on the spatialhypersurface
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
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Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
8182019 Luca Bai Otti
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8182019 Luca Bai Otti
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 73100
In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
8182019 Luca Bai Otti
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 75100
A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
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8182019 Luca Bai Otti
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Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
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M d i
8182019 Luca Bai Otti
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
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W f l i EOS
8182019 Luca Bai Otti
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Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
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Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
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Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
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Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
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We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
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Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
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Similarly with a mixed time-space projection of the left-hand-
side of the Einstein equations we obtain
Doing the same for the right-hand-side using the contracted
Codazzi equations and the definition of the momentum densitywe reach the equations called the momentum constraintequations
which are also 3 elliptic equations
The 4 constraint equations are the necessary and sufficientintegrability conditions for the embedding of the spacelikehypersurfaces in the 4D spacetime
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we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
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8182019 Luca Bai Otti
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We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
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In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
8182019 Luca Bai Otti
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As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
8182019 Luca Bai Otti
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
8182019 Luca Bai Otti
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We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
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First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
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where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
8182019 Luca Bai Otti
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
8182019 Luca Bai Otti
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
8182019 Luca Bai Otti
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
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Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
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8182019 Luca Bai Otti
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
8182019 Luca Bai Otti
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 75100
A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 44100
we must ensure that when going from one hypersurface at time to another at time all the vectors originating on endup on we must land on a single hypersurface
t+ δ t
tΣ1
2 1
Σ2
The most general of suchvectors that connect twohypersurfaces is
where is any spatial ldquoshiftrdquo vector Indeed we see that
so that the change in along is and so it is thesame for all points which consequently end up all on the samehypersurface
t tmicro
δ t = tmicronablamicrot = 1
Before proceeding to the derivation of the evolution equations
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 45100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 46100
We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 47100
In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 48100
As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 49100
Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 50100
We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 51100
We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 52100
First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 53100
NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
8182019 Luca Bai Otti
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
8182019 Luca Bai Otti
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 56100
Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 57100
iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
8182019 Luca Bai Otti
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 59100
where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
8182019 Luca Bai Otti
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
8182019 Luca Bai Otti
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
8182019 Luca Bai Otti
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
8182019 Luca Bai Otti
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
8182019 Luca Bai Otti
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
8182019 Luca Bai Otti
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
8182019 Luca Bai Otti
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
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Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
8182019 Luca Bai Otti
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8182019 Luca Bai Otti
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
8182019 Luca Bai Otti
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 45100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 46100
We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 47100
In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 48100
As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 49100
Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 50100
We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 51100
We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 52100
First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
8182019 Luca Bai Otti
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NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
8182019 Luca Bai Otti
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Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
8182019 Luca Bai Otti
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Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
8182019 Luca Bai Otti
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Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
8182019 Luca Bai Otti
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iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
8182019 Luca Bai Otti
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A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 59100
where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
8182019 Luca Bai Otti
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In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
8182019 Luca Bai Otti
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There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
8182019 Luca Bai Otti
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The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
8182019 Luca Bai Otti
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Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
8182019 Luca Bai Otti
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We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
8182019 Luca Bai Otti
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Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
8182019 Luca Bai Otti
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where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
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conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
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Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
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Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
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In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
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Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
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In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
8182019 Luca Bai Otti
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The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
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A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
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ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
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The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
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T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
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Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
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Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
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is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
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Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
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I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
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bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
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BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
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Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 46100
We can now express the last piece of the 3+1 decomposition andso derive the evolution part of the Einstein equations
As for the constraints we need suitable projections of the two sidesof the Einstein equations and in particular the two spatial ones
Using the Ricci equations one then obtains
where
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 47100
In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 48100
As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 49100
Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 50100
We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 51100
We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 52100
First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 53100
NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 54100
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 55100
Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 56100
Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 57100
iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 58100
A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 59100
where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 60100
In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 61100
There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 62100
The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 63100
Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 64100
We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 65100
Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 66100
where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 67100
conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 68100
Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 69100
Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 70100
In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 72100
Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 73100
In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 74100
The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 75100
A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 76100
ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 78100
T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 79100
Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 80100
Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 47100
In the spirit of the 3+1 formalism the natural choice for thecoordinate unit vectors is
i) three purely spatial coordinates with unit vectors
So far the treatment has been coordinate independent but in order to write computer programs we have to specify a coordinate basisDoing so can also be useful to simplify equations and to highlight
the ldquospatialrdquo nature of and
ii) one coordinate unit vector along the vector
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 48100
As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 49100
Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 50100
We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 51100
We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 52100
First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 53100
NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 54100
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 55100
Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 56100
Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 57100
iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 58100
A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 59100
where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 60100
In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 61100
There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 62100
The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 63100
Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 64100
We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 65100
Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 66100
where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 67100
conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 68100
Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 69100
Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 70100
In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 72100
Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 73100
In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 74100
The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 75100
A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 76100
ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 78100
T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 79100
Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 80100
Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
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V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
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Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 48100
As a result
ie the Lie derivative along is a simple partial derivative
ie the space covariant components of a timelike vector arezero only the time component is different from zero
ie the zeroth contravariant component of a spacelike vectorare zero only the space components are nonzero
Putting things together and bearing in mind that
8182019 Luca Bai Otti
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Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 50100
We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 51100
We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 52100
First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 53100
NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 54100
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 55100
Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 56100
Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 57100
iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 58100
A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 59100
where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 60100
In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 61100
There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 62100
The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 63100
Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 64100
We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 65100
Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 66100
where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 67100
conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 68100
Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 69100
Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 70100
In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 72100
Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 73100
In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 74100
The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 75100
A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 76100
ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 78100
T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 79100
Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 80100
Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 49100
Recalling that the spatial components of the 4D metricare the components of the 3D metric ( ) and
that (true in general for any spatial tensor) thecontravariant components of the metricin a 3+1 split are
Similarly since the covariant components are
Note that (ie are inverses) and thus theycan be used to raiselower the indices of spatial tensors
ij= γ
ij
γ
α0= 0
gmicroν
= γ microν
minus nmicron
ν
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 50100
We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 51100
We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 52100
First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 53100
NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 54100
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 55100
Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 56100
Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 57100
iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 58100
A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 59100
where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 60100
In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 61100
There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 62100
The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 63100
Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 64100
We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 65100
Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 66100
where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 67100
conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 68100
Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 69100
Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 70100
In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 72100
Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 73100
In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 74100
The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 75100
A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 76100
ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 78100
T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 79100
Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 80100
Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 50100
We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 51100
We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 52100
First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 53100
NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 54100
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 55100
Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 56100
Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 57100
iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 58100
A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 59100
where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 60100
In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 61100
There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 62100
The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 63100
Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 64100
We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 65100
Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 66100
where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 67100
conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 68100
Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 69100
Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 70100
In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 72100
Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 73100
In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 74100
The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 75100
A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 76100
ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 78100
T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 79100
Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 80100
Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 51100
We can now have a more intuitive interpretation of the lapseshift and spatial metric Using the expression for the 4D covariantmetric the line element is given by
It is now clearer that
bull the lapse measures proper time between two adjacent hypersurfaces
bull the shift relates spatial coordinatesbetween two adjacent hypersurfaces
bull the spatial metric measures distances between points onevery hypersurface
normal linecoordinate line
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 52100
First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 53100
NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 54100
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 55100
Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 56100
Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 57100
iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 58100
A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 59100
where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 60100
In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 61100
There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 62100
The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 63100
Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 64100
We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 65100
Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 66100
where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 67100
conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 68100
Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 69100
Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 70100
In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 72100
Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 73100
In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 74100
The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 75100
A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 76100
ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 78100
T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 79100
Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 80100
Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 52100
First step foliate the 4D spacetime in 3D spacelike hypersurfacesleveled by a scalar function the time coordinate This determinesa normal unit vector to the hypersurfaces
Second step decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric
Third step rewrite Einstein equations using such decomposed tensors Also selecting two functions the lapse and the shift that tell how
to relate coordinates between two slices the lapse measures the proper
time while the shift measures changes in the spatial coordinates
Fourth step select a coordinate basis and express all equationsin 3+1 form
The 3+1 or ADM (Arnowitt Deser Misner) formulation
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 53100
NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 54100
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 55100
Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 56100
Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 57100
iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 58100
A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 59100
where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 60100
In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 61100
There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 62100
The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 63100
Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 64100
We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 65100
Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 66100
where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 67100
conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 68100
Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 69100
Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 70100
In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 72100
Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 73100
In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 74100
The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 75100
A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 76100
ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 78100
T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 79100
Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 80100
Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 53100
NOTE the lapse and shift are not solutions of the Einsteinequations but represent our ldquogauge freedomrdquo namely thefreedom (arbitrariness) in which we choose to foliate thespacetime
Any prescribed choice for the lapse is usually referred to as ardquoslicing conditionrdquo while any choice for the shift is usuallyreferred to as rdquospatial gauge conditionrdquo
While there are infinite possible choices not all of them are
equally useful to carry out numerical simulations Indeed there is a whole branch of numerical relativity that isdedicated to finding suitable gauge conditions
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 54100
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 55100
Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 56100
Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 57100
iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 58100
A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 59100
where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 60100
In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 61100
There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 62100
The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 63100
Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 64100
We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 65100
Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 66100
where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 67100
conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 68100
Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 69100
Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 70100
In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 72100
Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 73100
In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 74100
The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 75100
A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 76100
ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 78100
T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 79100
Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 80100
Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 54100
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 55100
Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 56100
Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 57100
iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 58100
A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 59100
where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 60100
In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 61100
There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 62100
The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 63100
Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 64100
We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 65100
Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 66100
where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 67100
conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 68100
Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 69100
Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 70100
In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 72100
Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 73100
In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 74100
The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 75100
A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 76100
ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 78100
T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 79100
Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 80100
Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 55100
Suppose you want to follow the gravitational
collapse to a black hole and assume a simplisticgauge choice (geodesic slicing)
That would lead to a code crash as soon as asingularity forms No chance of measuring gws
One needs to use smarter temporal gauges
In particular we want time to progress atdifferent rates at different positions in thegrid ldquosingularity avoiding slicingrdquo (eg maximal
slicing)
Some chance of measuring gravitational waves
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 56100
Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 57100
iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 58100
A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 59100
where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 60100
In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 61100
There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 62100
The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 63100
Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 64100
We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 65100
Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 66100
where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 67100
conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 68100
Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 69100
Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 70100
In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 72100
Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 73100
In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 74100
The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 75100
A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 76100
ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 78100
T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 79100
Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 80100
Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 56100
Good idea mathematically but unfortunately this leads to ellipticequations which are computationally too expensive to solve ateach time
ii) fix the lapse and shift by requiring they satisfy some
condition eg maximal slicing for the lapse
which has the desired ldquosingularity-avoidingrdquo properties
Different recipes for selecting lapse and shift are possible
i) make a guess (ie prescribe a functional form) for the lapseand shift eg geodesic slicing
obviously not a good idea
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 57100
iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 58100
A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 59100
where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 60100
In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 61100
There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 62100
The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 63100
Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 64100
We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 65100
Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 66100
where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 67100
conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 68100
Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 69100
Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 70100
In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 72100
Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 73100
In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 74100
The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 75100
A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 76100
ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 78100
T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 79100
Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 80100
Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 57100
iii) determine the lapse and shift dynamically by requiring that they satisfy comparatively simple evolution equations
This is the common solution The advantage is that theequations for the lapse and shift are simple time evolutionequations
A family of slicing conditions that works very well to obtain
both a strongly hyperbolic evolution equations and stablenumerical evolutions is the Bona-Masso slicing
where and is a positive but otherwisearbitrary function
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 58100
A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 59100
where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 60100
In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 61100
There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 62100
The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 63100
Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 64100
We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 65100
Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 66100
where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 67100
conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 68100
Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 69100
Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 70100
In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 72100
Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 73100
In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 74100
The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 75100
A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 76100
ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 78100
T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 79100
Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 80100
Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 58100
A high value of the metric
components means that thedistance between numerical grid
points is actually large and this
causes problems
In addition large gradients my benumerically a problem
Choosing a ldquobadrdquo shift may even
lead to coordinates singularities
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 59100
where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 60100
In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 61100
There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 62100
The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 63100
Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 64100
We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 65100
Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 66100
where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 67100
conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 68100
Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 69100
Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 70100
In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 72100
Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 73100
In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 74100
The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 75100
A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 76100
ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 78100
T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 79100
Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 80100
Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 59100
where and acts as a restoring force to avoid largeoscillations in the shift and the driver tends to keep theGammas constant
A popular choice for the shift is the hyperbolic ldquoGamma-driverrdquo condition
Overall the ldquo1+logrdquo slicing condition and the ldquoGamma-
driverrdquo shift condition are the most widely used both invacuum and non-vacuum spacetimes
B
i
equiv part tβ i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 60100
In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 61100
There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 62100
The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 63100
Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 64100
We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 65100
Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 66100
where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 67100
conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 68100
Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 69100
Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 70100
In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 72100
Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 73100
In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 74100
The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 75100
A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 76100
ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 78100
T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 79100
Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 80100
Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 60100
In practice the actual
e x c i s i o n r e g i o n i s a
ldquolegosphererdquo (black region) and
is placed well inside theapparent horizon (which is
found at every time step) and
is allowed to move on the grid
apparent horizon
The region of spacetime inside a horizon
( yellow region) is causally disconnected
from the outside (blue region)
So a region inside a horizon may be
excised from the numerical domain
This is successfully done in pure
spacetime evolutions since the work
of Nadeumlzhin Novikov Polnarev (1978)
Baiotti et al [PRD 71 104006
(2005)] and other groups [Duez et al
PRD 69 104016 (2004)] have shown
that it can be done also in non-vacuum
simulations
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 61100
There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 62100
The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 63100
Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 64100
We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 65100
Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 66100
where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 67100
conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 68100
Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 69100
Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 70100
In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 72100
Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 73100
In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 74100
The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 75100
A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 76100
ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 78100
T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 79100
Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 80100
Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 61100
There is an alternative to explicit excision Proposed independently by Campanelli et
al PRL96 111101 (2006) and Baker et al PRL96 111102 (2006) it is nowadays a very
popular method for moving--black-hole evolution
It consists in using coordinates that allow the punctures (locations of the
singularities) to move through the grid but do not allow any evolution at the puncture
point itself (ie the lapse is forced to go to zero at the puncture though not the
shift vector hence the ldquofrozenrdquo puncture can be advected through the domain) The
conditions that have so far proven successful are modifications to the so-called 1+log
slicing and Gamma-driver shift conditions
In practice such gauges take care that the punctures are never located at a grid
point so that actually no infinity is present on the grid This method has proven to be
stable convergent and successful A few such gauge options are available with
parameters allowed to vary in determined ranges (but no fine tuning is necessary)This mechanism works because it implements an effective excision (ldquoexcision without
excisionrdquo) It has been shown that in the coordinates implied by the employed gauges
the singularity is actually always outside the numerical domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 62100
The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 63100
Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 64100
We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 65100
Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 66100
where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 67100
conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 68100
Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 69100
Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 70100
In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 72100
Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 73100
In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 74100
The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 75100
A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 76100
ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 78100
T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 79100
Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 80100
Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 62100
The ADM equations are perfectly all right mathematically butnot in a form that is well suited for numerical implementation
Indeed the system can be shown to be weakly hyperbolic (namely its set of eigenvectors is not complete) and hence ldquoill-
posedrdquo (namely as time progresses some norm of the solutiongrows more than exponentially in time)
In practice numerical instabilities rapidly appear that destroy the solution exponentially
However the stability properties of numerical implementationscan be improved by introducing certain new auxiliary functionsand rewriting the ADM equations in terms of these functions
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 63100
Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 64100
We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 65100
Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 66100
where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 67100
conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 68100
Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 69100
Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 70100
In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 72100
Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 73100
In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 74100
The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 75100
A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 76100
ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 78100
T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 79100
Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 80100
Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 63100
Instead we would like to have only terms of the type
In this equation there are mixed second derivatives sincecontains mixed derivatives in addition to a Laplace operatoracting on
Think of the above system as a second-order system for
because without the mixed derivatives the 3+1 ADMequations could be written in a such way that they behavelike a wave equation for
Letrsquos inspect the 3+1 evolution equations again
γ ij
part micropart microγ ij
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 64100
We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 65100
Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 66100
where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 67100
conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 68100
Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 69100
Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 70100
In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 72100
Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 73100
In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 74100
The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 75100
A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 76100
ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 78100
T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 79100
Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 80100
Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 64100
We care to have wave equations like
because wave equations are manifestly hyperbolic andmathematical theorems guarantee the existence anduniqueness of the solutions (as seen in previous lectures of
this school)
We can make the equations manifestly hyperbolic indifferent ways including
i) using a clever specific gauge
ii) introducing new variables which follow additional equations
(imposed by the original system)
These methods aim at removing the mixed-derivative term
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 65100
Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 66100
where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 67100
conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 68100
Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 69100
Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 70100
In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 72100
Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 73100
In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 74100
The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 75100
A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 76100
ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 78100
T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 79100
Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 80100
Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 65100
Let choose a clever gauge that makes the ADM equationsstrongly hyperbolic
The generalized harmonic formulation is based on ageneralization of the harmonic coordinates
When such condition is enforced in the Einstein equations the principal part of the equations for each metric element
becomes a scalar wave equation with all nonlinearities andcouplings between the equations relegated to lower order terms
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 66100
where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 67100
conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 68100
Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 69100
Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 70100
In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 72100
Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 73100
In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 74100
The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 75100
A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 76100
ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 78100
T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 79100
Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 80100
Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 66100
where the a are a set of source functions
The and the equations for their evolution must be suitably
chosen
The harmonic condition is known to suffer frompathologies
However alternatives can be found that do not suffer fromsuch pathologies and still have the desired properties of theharmonic formulation In particular the generalized harmoniccoordinates have the form
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 67100
conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 68100
Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 69100
Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 70100
In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 72100
Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 73100
In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 74100
The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 75100
A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 76100
ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 78100
T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 79100
Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 80100
Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 67100
conformal factor
conformal 3-metric trace of extrinsic curvature
trace-free conformalextrinsic curvature
ldquoGammasrdquo
φ
˜γ ij
K
Aij
˜Γ
i
New evolution variables are introduced to obtain from the
ADM system a set of equations that is strongly hyperbolicA successful set of new evolution variables is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 68100
Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 69100
Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 70100
In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 72100
Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 73100
In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 74100
The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 75100
A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 76100
ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 78100
T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 79100
Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 80100
Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 68100
Dtγ ij = minus2α Aij
Dtφ = minus1
6αK
Dt Aij = eminus4φ [minusnablainablajα + α (Rij minus S ij)]TF + α
K Aij minus 2 Ail
Alj
DtK = minusγ ijnablainablajα + α
Aij Aij + 1
3K 2 + 1
2 (ρ + S )
DtΓi = minus2 Aijpart jα + 2α
ΓijkAkj
minus 2
3γ ijpart jK minus γ ijS j + 6 Aijpart jφ
minuspart jβ l
part lγ ij
minus 2γ m(j
part mβ i)
+
2
3 γ ij
part lβ l
Dt equiv part t minus Lβwhere
These equations are also known as the BSSN-NOK equations or moresimply as the conformal traceless formulation of the Einstein equations
And the ADM equations are then rewritten as
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 69100
Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 70100
In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 72100
Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 73100
In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 74100
The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 75100
A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 76100
ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 78100
T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 79100
Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 80100
Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 69100
Although not self evident the BSSN-NOK equations arestrongly hyperbolic with a structure which is resembling the
1st-order in time 2nd-order in space formulation
scalar wave equation
conformal tracelessformulation
The BSSN-NOK equations are nowadays the most widelyused form of the Einstein equations and have demonstrated
to lead to stable and accurate evolution of vacuum (binary--black-holes) and non-vacuum (neutron-stars) spacetimes
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 70100
In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 72100
Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 73100
In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 74100
The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 75100
A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 76100
ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 78100
T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 79100
Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 80100
Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 70100
In addition to the (6+6+3+1+1=17) hyperbolic evolution equations
to be solved from one time slice to the next there are the usual3+1=4 elliptic constraint equations
and 5 additional constraints are introduced by the new variables
NOTE most often these equations are not solved but only monitored to verify that
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 72100
Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 73100
In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 74100
The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 75100
A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 76100
ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 78100
T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 79100
Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 80100
Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 71100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 72100
Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 73100
In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 74100
The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 75100
A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 76100
ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 78100
T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 79100
Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 80100
Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 72100
Computing the waveforms is the ultimate goal of a large
portion of numerical relativityThere are several ways of extracting GWs from numericalrelativity codes the most widely used of which are
Both are based of finding some gauge invariant quantities or
the perturbations of some gauge-invariant quantity and torelate them to the gravitational waveform
bull Weyl scalars (a set of five complex scalar quantities describing the curvature of a 4D spacetime)
bullperturbative matching to a Schwarzschild background
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 73100
In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 74100
The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 75100
A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 76100
ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 78100
T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 79100
Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 80100
Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 73100
In both approaches ldquoobserversrdquo are placed on nested 2-spheres
and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate thegauge-invariant perturbations of a Schwarzschild black hole
Once the waveforms
are calculated all therelated quantities theradiated energymomentum andangular momentum canbe derived simply
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 74100
The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 75100
A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 76100
ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 78100
T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 79100
Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 80100
Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 74100
The ADM equations are ill posed and not suitable for
numerical integrations
Alternative formulations (BSSN-NOK GHC) have beendeveloped and shown to be strongly hyperbolic and hence well-posed and effective to solve Einstein equations
Getting a good formulation of the Einstein equations willwork only in conjunction with good gauge conditionsldquo1+logrdquoslicing ldquoGamma-driverrdquo conditions work well in a number ofconditions
Any astrophysical prediction needs the calculation ofldquorealisticrdquo initial data and hence the solution of elliptic equations
Gravitational waves are present in simulations and can beextracted with great accuracy
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 75100
A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 76100
ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 78100
T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 79100
Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 80100
Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 75100
A first course in general relativity Bernard Schutz Cambridge University Press (2009)
Gravitation C Misner K Thorn JA Wheeler Ed W H Freeman (1973)
Numerical Relativity Thomas Baumgarte and Stuart ShapiroCambridge University Press (2010)
Introduction to 3+1 Numerical Relativity Miguel AlcubierreOxford University Press (2008)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 76100
ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 78100
T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 79100
Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 80100
Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 76100
ie 4 conservation equations for the energy and momentum
ie conservation of baryon number and equation of state (EOS)
(microphysics input)
The additional set of equations to solve is
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 78100
T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 79100
Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 80100
Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 77100
The evolution of the magnetic field obeys Maxwellrsquos equations
part
part t
radic γ B
= nablatimes
α v minus β
timesradic
γ B
that give the divergence-free condition
and the equations for the evolution of the magnetic field
nablaν
lowastF microν
= 0
nablamiddot radic γ
B = 0
for a perfect fluid with infinite conductivity (ideal MHD approximation)
lowastF microν
=1
2microνσρ
F σρ
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 78100
T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 79100
Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 80100
Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 78100
T microν = (ρ + ρε + b2)umicrouν + p + 1
2
b2 gmicroν
minus bmicrobν
where
is the rest-mass density
is the specific internal energy
u is the four-velocity
p is the gas pressure
vi is the Eulerian three-velocity of the fluid (Valencia formulation)
W the Lorentz factor
b the four-vector of the magnetic field
Bi the three-vector of the magnetic field measured by an Eulerian observer
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 79100
Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 80100
Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 79100
Letrsquos recall the equations we are dealing with
This is not yet ldquorealrdquoastrophysics but ourapproximation toldquorealityrdquo
Still very crude butit can be improvedmicrophysics for theEOS viscosityradiation transportnabla
lowast
ν F microν = 0 (Maxwell eqs induction zero div)
The complete set of equations is solved with theCactusCarpetWhisky codes
+1 + 3 + 1
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 80100
Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 80100
Cactus (wwwcactuscodeorg) is a computational
ldquotoolkitrdquo developed at the AEICCT and provides a general
infrastructure for the solution in 3D and on parallel
computers of PDEs (eg Einstein equations)
Whisky (wwwwhiskycodeorg) developed at the AEI
Osaka University Southampton LSU for the solution of t h e r e l a t i v i s t i c h y d r o d y n a m i c s a n dmagnetohydrodynamics equations in arbitrarily curvedspacetimes
Carpet (wwwcarpetcodeorg) provides box-in-box
adaptive mesh refinement with vertex-centered grids
CactusCarpetWhisky
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 81100
is a driver mainly developed by E
Schnetter [CQG 21 1465 (2004)] which has removed the limitation ofusing uniform 3D grids
Carpet follows a (simplified) Berger-Oliger
[J Comput Phys 53 484 (1984)] approach to
mesh refinement that is
bull refined subdomains consist of a set of cuboid (=rectangular parallelepiped) grids
bull refined subdomains have boundaries alignedwith the grid lines
bull the refinement ratio between refinement levels
is constant
While the refined meshes are not automatically moving on the grid they can beactivated and deactivated during the evolution obtaining a progressive fixed meshrefinement or even a moving-grid mesh refinement
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 82100
Many codes for numerical relativity are publicly available from the recent project of
the which aims at providing computational toolsfor the community
It includes
bull spacetime evolution code
bull GRHydro code (not completely tested yet see also the Whisky webpage)
bull GRMHD code (in the near future)
bull mesh refinement
bull portability
bull simulation factory(an instrument to managesimulations)
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 83100
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 84100
Definition in general for a hyperbolic system of equations a
Riemann problem is an initial-value problem with initial condition
given by
where UL and UR are two constant vectors representing the ldquoleftrdquo
and ldquorightrdquo state
For hydrodynamics a (physical) Riemann problem is the evolution of
a fluid initially composed of two states with different and constant
values of velocity pressure and density
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 85100
I n HR SC me th od s e a chd i s c r e t i s a t i o n - p r o d u c e ddiscontinuity is considered a local
Riemann problem
There are powerful numericalmethods to solve Riemann
problems both exactly andapproximately
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 86100
bull High-order (up to 8th) finite-difference techniques for the field equations
bull Flux conservative form of HD and MHD equations with constraint transport
or hyperbolic divergence-cleaning or evolution of the vector potential for themagnetic field HRSC methods with different solvers (HLLE Roe Marquina) andreconstruction methods (linear PPM)
bull Multiple options for the wave extraction ( Weyl scalars gauge-invariant pertbs)
bull AMR with moving grids
bull Accurate measurements of BH properties through apparent horizons (IH)
bullUse excision (matter andor fields) if needed
bull Idealized (analytic) EoSs (work in progress to include more detailed EoSs)
bull Single-fluid description no superfluids nor crusts
bull Only inviscid fluid so far (not necessarily bad approximation)
bull Radiation and neutrino transport totally neglected (work in progress)
bull Very coarse resolution far from regimes where turbulencedynamos develop
BINARY NEUTRON STAR MERGERS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 87100
BINARY NEUTRON-STAR MERGERS
bull Baiotti Giacomazzo Rezzolla 2008 PRD 78 084033bull Baiotti Giacomazzo Rezzolla 2009 CQG 26 114005
bull Giacomazzo Rezzolla Baiotti 2009 MNRAS 399 L164-L168
bull Rezzolla Baiotti Giacomazzo Link Font 2010 CQG 27 114105
THE INITIAL GRID SETUP FORTHE BINARY
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 88100
V i s u a l i z a t i o n b y G i a c o m a z z o
K a e h l e r R e z z o l l a
THE INITIAL GRID SETUP FOR THE BINARY MASS 16X2 IDEAL-FLUID EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 89100
M d i
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 90100
Matter dynamicshigh-mass binary
soon after the merger the torus isformed and undergoes oscillations
Merger
Collapse toBH
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 91100
W f l i EOS
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 92100
Waveforms polytropic EOShigh-mass binary
The full signal from the inspiral to theformation of a BH has been computed
Merger Collapse to BH
I i t f th EOS id l fl id l i E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 93100
Imprint of the EOS ideal-fluid vs polytropic EoS
After the merger a BH is producedover a timescale comparable with the
dynamical one
After the merger a BH is producedover a timescale larger or much
larger than the dynamical one
W f id l fl id E S
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 94100
Waveforms ideal-fluid EoS
low-mass binary
C ti ti f th d
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 95100
Conservation properties of the code
conservation of energy
low-mass binary
conservation of angular momentum
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 96100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from themerger
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 97100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thebar-deformed HMNS
Imprint of the EOS frequency domain
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 98100
Imprint of the EOS frequency domain
The pre-merger dynamics are verysimilar the post-merger phase isvery different
Contributions from thecollapse to BH
E t di th k t MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 99100
We have considered the same models also when an initially
poloidal magnetic field of ~108
or ~1012
G is introduced
The magnetic field is added by hand using the vector potential
where and are two constants definingrespectively the strength and the extension of the magnetic fieldinside the star n=2 defines the profile of the initial magnetic field
The initial magnetic fields are therefore fully contained inside the
stars ie no magnetospheric effects
Ab P cut = 004timesmax(P )
Aφ = Abr2[max(P minus P cut 0)]
n
Extending the work to MHD
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary
8182019 Luca Bai Otti
httpslidepdfcomreaderfullluca-bai-otti 100100
Idealminus
fluid M = 165 M B = 10
12
G
Note that the torus is much less dense and a large
plasma outflow is starting to be launchedValidations are needed to confirm these results
Typical evolution for a magnetised binary