1. 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

2. 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

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

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

5. 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)

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

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

8. String theory
Photon
Graviton
Massive particle

9. D-branes

10. Dualities

11. Dualities

12. Dualities
SBH S =

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

14. 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’

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

16. Eternal black holes

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

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

19. Entanglement entropy Results q≠0: Results: 50% ↑ 50% ↑ 50% ↓ 50% ↓ q 12 2
Results:
50% ↑
50% ↓
Results q≠0:
50% ↑
50% ↓

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

21. The vacuum state
|0 t x

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

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

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