gbr/GW/Lect1.pdf · 1 −z k 2 Free test-body trajectories depend only on initial position,...
Transcript of gbr/GW/Lect1.pdf · 1 −z k 2 Free test-body trajectories depend only on initial position,...
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Gravitational wave generation, propagation and detection
The science enabled by gravitational wave observations
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1 ∼(v
c
)2∼ GM
c2L∼ φG
VirialThm
Energetic
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No (negligible) reddening
No (negligible) extinction
Undistorted view of central engine dynamics
Charge? Mass
Current? Momentum
Radiation? Time varying mass, current distributions
No gravitational wave zone of avoidance!
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∂2
∂t2
(zj1 − zj2
) ( ∂2φG∂xj∂xk
) (zk1 − zk2
)
Free test-body trajectories depend only on initial position, velocity
Can’t distinguish gravity from coordinate system acceleration on basis of single trajectory
“Tidal force” acting on separation between nearby trajectories
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gμν = ημν + hμν
ḧjk ∼ ∂2φG
∂xj∂xk
ds2 = gμνdxμdxν
∂2
∂t2
(zj1 − zj2
) ( ∂2φG∂xj∂xk
) (zk1 − zk2
)
Space-time interval
Metric
Infinitesimal coordinate separation between space-time points
Minkowski metric
Metric perturbation
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�h̄μν = −16πGc2
Tμν
∂
∂xμh̄μν = 0
(h̄μν = hμν − 12ημνh
h = ηαβhαβ)
�Aμ = −4πJμ∂Aμ
∂xμ= 0
gμν = ημν + hμν
hμνAμ
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�Aμ = −4πJμ
Aμ(t,x) = −4π∫
d3x′Jμ(t − |x−x′|c ,x′)
|x − x′|
1|x|
[Jμ(x′) +
x′j
cJ̇μ(x′) + . . .
]ret
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Ijk =∫
d3x
[xjxk − x
2
3δjk
]ρ(�x)
hTTjk =2Gc2r
[Ïjk
]TTret
“Transverse-Traceless” gauge specialization. Project transverse to direction of wave propagation and remove trace
Trace-free quadrupole moment of matter distribution
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L =1
32Gcπ
∫d2Ωr2ḣTTjk ḣ
TTjk
=15
G
c5...I jk
...I jk
∼ Gc5
(Mv2
T
)2∼ K.E./T
c5/G
K.E.T
h ∼ 10−39 1 kmr
M
1000 kg
(v
300 m/s
)
c5
G= 3.6 × 1059 erg
s
forb~125 Hz
h ∼ 10−23 100 Mpcr
M
3M�150 km
R
(!)
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For radiation propagating along z axis...
Ixx =Mr2
4cos2 ωt
Ixy =Mr2
4cos ωt sinωt
Iyy =Mr2
4sin2 ωt
Ixx =Mr2
4
[cos2 ωt − 1
3
]
Ixy =Mr2
4cos ωt sinωt
Iyy =Mr2
4
[sin2 ωt − 1
3
]
Izz = −Mr2
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hxx = −hyy = −GM (ωr)2
c2Rcos 2ωt
hxy = hyx = −GM (ωr)2
c2Rsin 2ωt
Ïxx = −M2 (ωr)2 cos 2ωt
Ïxy = −M2 (ωr)2 sin 2ωt
Ïyy =M
2(ωr)2 cos 2ωt
L =2G5c5
(ωr)4 (ωM)2
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S. L. Shapiro, S. A. Teukolsky. 1983. Black Holes, White Dwarfs and Neutron Stars (Wiley: New York)
P. C. Peters, J. Mathews. 1963. Gravitational radiation from point masses in a Keplerian orbit. Phys. Rev. 131:435
T. X. Thuan, J. P. Ostriker. 1974. Gravitational radiation from stellar collapse. Ap. J. Lett. 191:L105 – L107.
C. Van Den Broeck. 2005. The gravitational wave spectrum of non-axisymmetric, freely precessing neutron stars. Class. Quant. Grav. 22:1825 – 1839.
What is the radiation from a rapidly rotating, non-axisymmetric, non-precessing neutron star?
What is the braking index of for a pulsar whose spin-down is dominated by gravitational waves?
Consider a circular orbit binary system, losing energy adiabatically to gravitational waves. How does the binary period change with time?
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