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Black Hole Entropy Black holes – a review Black holes in string theory
Amos Yarom Black holes – a review Black holes in string theory Entanglement and black holes Entanglement in string theory
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Black holes R≤ Rs=2GM/c2 A wrong derivation yielding correct results:
If nothing can escape then: Yielding: Black hole condition R≤ Rs=2GM/c2
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The event horizon Singularity Singularity formed formed
x y t Event Horizon formed Schwartzshield radius Event Horizon formed x y Schwartzshield radius
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Black hole thermodynamics
J. Beckenstein (1973) S. Hawking (1975) S =0 S A S = ¼ A TH=1/(8pM)
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What is entropy? S=k·ln(N) S=-k Tr(rlnr) Macroscopic state
Microscopic states |☺☺O> |☺O☺> | O ☺☺> S=-k 3 1/3 ln 1/3 = k ln 3 S=k ln 3 S=-k Tr(rlnr)
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What does black hole entropy mean?
x p ? x ?
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String theory l Xm(t0,s) Xm(t,s0) Xm(t0,s0) l0 x y t Xm (l0) t s
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String theory Photon Graviton Massive particle
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D-branes
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Dualities
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Dualities
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Dualities SBH S =
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An explicit example: AdS/CFT
Maldacena (1997) AdS space CFT Minkowski space Anti deSitter deSitter
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Semiclassical gravity:
AdS BH Entropy S. S. Gubser, I. R. Klebanov, and A. W. Peet (1996) Anti deSitter +BH CFT , T>0 ? AdS/CFT S/A 1/R SBH=A/4 S=A/3 Free theory: l 0 Semiclassical gravity: R>>a’
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Eternal black holes Singularity formed Schwartzshield radius t t
x y t x y t Event Horizon formed Schwartzshield radius
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Eternal black holes
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Eternal Black holes r=0 t t=0 x
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Eternal Black holes r=0 t t=0 x
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Entanglement entropy Results q≠0: Results: 50% ↑ 50% ↑ 50% ↓ 50% ↓ q 1
2 2 Results: 50% ↑ 50% ↓ Results q≠0: 50% ↑ 50% ↓
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Entanglement entropy S=0 All |↓22↓| S1=Trace (r1lnr1)=ln2 elements 1
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The vacuum state |0 t x
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AdS BH Maldacena (2003) AdS BH CFTCFT, T=0 CFT, T>0 ? AdS/CFT |0
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Generalization |0 BH spacetime Field theory
R. Brustein, M. Einhorn and A.Y. (2005) BH spacetime Field theory |0
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Summary BH’s have entropy.
String theory may evaluate this entropy explicitly. This evaluation is consistent with entanglement entropy.
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