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Black Hole Entropy

Black Hole Entropy. Amos Yarom. Black holes – a review Black holes in string theory Entanglement and black holes Entanglement in string theory. A wrong derivation yielding correct results:. If nothing can escape then:. Yielding:. Black holes. Black hole condition. R≤. R s =2GM/c 2.

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Black Hole Entropy

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  1. Black Hole Entropy Amos Yarom • Black holes – a review • Black holes in string theory • Entanglement and black holes • Entanglement in string theory

  2. A wrong derivation yielding correct results: If nothing can escape then: Yielding: Black holes Black hole condition R≤ Rs=2GM/c2

  3. Singularity formed Singularity formed t Event Horizon formed Event Horizon formed Schwartzshield radius y y x x Schwartzshield radius The event horizon

  4. Black hole thermodynamics S. Hawking (1975) J. Beckenstein (1973) S =0 S  A S = ¼ A TH=1/(8pM)

  5. Macroscopic state Microscopic states |☺☺O> |☺O☺> | O ☺☺> What is entropy? S=k·ln(N) S=-k 3 1/3 ln 1/3 = k ln 3 S=k ln 3 S=-k Tr(rlnr)

  6. What does black hole entropy mean? p x ? p ? x ?

  7. x t s y t String theory l Xm(t0,s0) Xm(t0,s) Xm(t,s0) l0 Xm (l0)

  8. String theory Photon Graviton Massive particle

  9. D-branes

  10. Dualities

  11. Dualities

  12. Dualities SBH S =

  13. Minkowski space Anti deSitter deSitter An explicit example: AdS/CFT Maldacena (1997) AdS space CFT

  14. Anti deSitter +BH CFT AdS/CFT S/A 1/R Free theory: l 0 Semiclassical gravity: R>>a’ AdS BH Entropy S. S. Gubser, I. R. Klebanov, and A. W. Peet (1996) , T>0 ? S=A/3 SBH=A/4

  15. Singularity formed t t Event Horizon formed y y x x Schwartzshield radius Eternal black holes

  16. Eternal black holes

  17. r=0 t Eternal Black holes t=0 x

  18. r=0 t Eternal Black holes t=0 x

  19. 1 1 2 2 q Entanglement entropy Results q≠0: 50% ↑ 50% ↓ Results: 50% ↑ 50% ↓

  20. All |↓22↓| elements 1 2 Entanglement entropy S=0 S1=Trace (r1lnr1)=ln2

  21. t x The vacuum state |0

  22. AdS/CFT AdS BH Maldacena (2003) AdS BH CFTCFT, T=0 CFT, T>0 ? |0

  23. Generalization R. Brustein, M. Einhorn and A.Y. (2005) Field theory BH spacetime |0

  24. Summary • BH’s have entropy. • String theory may evaluate this entropy explicitly. • This evaluation is consistent with entanglement entropy.

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