On Fuzzball conjecture
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Transcript of On Fuzzball conjecture
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On Fuzzball conjecture
Seiji Terashima (YITP, Kyoto)
based on the work (PRD78 064029(2008), arXiv:0805.1405)in collaboration with Noriaki Ogawa (YITP)
2009 Feb 19 at Kinosaki
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1. Introduction
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In general relativity (or classical gravity):
Black hole solutions exist and behave like blackbody:
Thermodynamics
d E = T d S
δS ≥ 0
Blackbody radiation
Black hole
d M = κd A
δA ≥ 0
Hawking radiationM: Mass of B.H.A: Area of horizonκ: surface gravity
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Irreversibility ↔ horizon.
No information inside the B.H.can escape outside the horizon.
Information loss problem(quantum theory should be unitary)
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Anyway, black hole is a classical solution,but thermodynamic object!
General relativity will be just an effective action,and there will be micro states for the black hole,perhaps quantum mechanically.
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String Theory- is well defined and understood perturbatively- is useful for Mathematics (ex. Mirror symmetry)- is applied to the QCD (ex. Holographic QCD)- can be applied to Particle Phenomenology and can be the Theory Of Everything.
- includes Quantum Gravity
Remember why String Theory is interesting:
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Indeed, in string theorywe can find the microstates of B.H.
and show:
log (# of microstates)= Area of B.H.
for some BPS B.H.Strominger-Vafa
using the Gauge/Gravity duality.
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a Puzzle remains:
Near the horizon, classical general relativity seems valid because curvature is small
andthe quantum effects would be confined within
the plank length near the singularity
Information loss? Inconsistency?
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Fuzzball conjecture:
“Black Hole” does not exist quantum mechanically (in string theory).
(There is no horizon in quantum gravity.)
Instead of black hole, somethings like fuzzball( 毛玉) exist.
Mathur and his collaborators
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Fuzzball(has complicated topololgy)
Black hole
Corse graining(粗視化)
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B.H. is macroscopic object andhas very large number of degree of freedom, N.
Quantum effects will spread out by the large N effect.
→ 1. Macroscopic “hozizon” appear2. Quantum effect near horizon !
Key point:
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In supergravity, some classical solutions without horizon and singularity
which are approximately same as the B.H. outside “horizon”were found (fuzzball solution).
→ “horizon” appear as approximate notion
Note 1:the fuzzball in general are not represented as
classical solution. Only for special cases.
an evidence:
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Note 2:
BPS properties are essential to find the solutions.
BPS equations is linear → reduced to one like Laplace equation
→ infinite sum of the solutions are also solutions→Fuzzball solutions
No hair “theorem” is terribly violated
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2. Half BPS Fuzzball in AdS5xS5
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AdS5/CFT4
AdS5 x S5 ↔ N=4 SU(N) SYM
restrict to ½ BPS sector
Lin-Lunin-Maldacena ↔ N free fermions in (LLM)geometry harmonic oscillator potential
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Other parts
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AdS5x S5 solution
In the coordinate , the metric becomes
AdS5xS5 geometry within the global coordinate
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[Note] fermion distribution can be defined for special class of state in Hilbert space, i.e. semi-classical or
“coherent states” spanning the base of Hilbert space
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• Pauli’s exclusion principle means for CFT side.
• LLM geometry have Closed Time like Curve for outside this region. Thus these solutions will be unphysical in quantum theory.
• LLM geometry is smooth for
• This phase space us quantized by
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Thus, in this ½ BPS sectors of AdS5xS5 example, all microstates are represented by smooth supergravity solutions, which are the fuzzballs!
More presicely, all states
of a basis of microstates
are represented
by supergravity solutions.
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Smooth geometry ↔ pure states in CFT
Interpretation of the singular geometries in LLMis Mixed/coarse grained states in CFT.
Singular geom. = “gray droplets”
Superstar=simplest gray droplets=uniform disk
Balasbramnian et.al.
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Suparstar has naked singularity without horizon,
but we expect it will have horizon after including quatum effects
5d N=2 supergravity reduced form is
BPS “black hole”, but not extremal.If the horizon is at ζ ,
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Summary of our work:“Coarse-graining of bubbling geometries
and the fuzzball conjecture”
• Consider charged “B.H.” called “Superstar” in AdS space-time
• Estimate the entropy of superstar, as log(# of microstates)• Estimate the "horizon" size of the superstar, based on the
fuzzball conjecture, from gravity side.
• We find the Bekenstein-Hawking entropy computed from this "horizon" agrees with log(# of microstates)
• Thus, this result supports the fuzzball conjecture.
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Entropy = log of number of states which are not distinguishable
from one another by macroscopic observations.
So, what is “macroscopic observer”?We assume that
One can only measure physical quantities up to
If metric perturbation is bigger than We can detect two geometries are different
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Consider two similar LLM geometries with fermion distributions and
and their difference
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Then, we can compute the entropy of the superstar (and other non-smooth geometries) by
counting the number of possible microstates as
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Now we observe region very close to the fuzzballand see where is “horizon” .
Here the “horizon” is the surface of the region where the typical microstates are different each other.
After some computation, we find “horizon” is at
or in the coordinate for the superstar.
Therefore, according to previous formula, we have
We take with very small α
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Conclusions
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• Summary– For some BPS black holes we find there are fuzzball solutions in
super gravity, which are smooth and has no horizon.– These will correspond to the microstates of B.H.
• Future directions – Including higher derivative corrections– Non BPS, non extremal
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Fin.