1. On black hole microstates Introduction BH entropy Entanglement entropy BH microstates Amos Yarom. Ram Brustein. Martin Einhorn.

2. Geometry 

3. General relativity G  =T  =0 r=2M r=0 Coordinate singularity Spacetime singularity

4. Coordinate singularities x y r  x=r cos  y=r sin 

5. Kruskal extension Previous coordinates: t x r=2M r=0 t=0 t=1/2 t=1 t=3/2 x

6. Kruskal extension t x r=2M r=0 

7. Black hole thermodynamics J. Beckenstein (1973) S. Hawking (1975) S  A T H =1/(8  M) S = ¼ A S =0

8. What does BH entropy mean? BH Microstates Horizon states Entanglement entropy

9. 1 2 Results: 50% ↑ 50% ↓ Results  ≠0: 50% ↑ 50% ↓ 1 2 

10. Entanglement entropy S=0 S  =Trace (   ln  1 )=ln2 S  =Trace (   ln  2 )=ln2 All |↓  22  ↓| elements 1 2

11. The vacuum state |0  t x r=0 r=2M

12. Finding  1  (x,0)=  (x) x t  ’(x)  ’’(x) Tr 2  (  ’  ’’   1 (  ’ 1,  ’’ 1 ) =  1  ’ 1  ’’ 1    Exp[-S E ] D   (x,0 + ) =  ’ 1 (x)  (x,0 - ) =  ’’ 1 (x)  (x,0 + ) =  ’ 1 (x)  2 (x)  (x,0 - ) =  ’’ 1 (x)  2 (x)   Exp[-S E ] D  D  2  (x,0 + )=  ’(x)  (x,0 - )=  ’’(x)  (x,0 + )=  ’(x)  (x,0 - )=  ’’(x)

13. What does BH entropy mean? BH Microstates Horizon states Entanglement entropy √ x t  ’ 1 (x)  ’’ 1 (x)  ’| e -  H |  ’’  Kabbat & Strassler (1994), R. Brustein, M. Einhorn and A.Y. (to appear) Finding  1  1  ’ 1  ’’ 1    Exp[-S E ] D   (x,0 + ) =  ’ 1 (x)  (x,0 - ) =  ’’ 1 (x)   

14. Counting of microstates (Conformal) field theory Curved spacetime Quantized gravity String theory

15. AdS/CFT AdS spaceCFT Minkowski space deSitter Anti deSitter  O Z(  b =  0 )  Exp(    O dV)  = Maldacena (1997)

16. S BH =A/4 S=A/3 Semiclassical gravity: R>>  ’ Free theory:  0 S/A 1/R AdS BH Entropy S. S. Gubser, I. R. Klebanov, and A. W. Peet (1996) Anti deSitter +BH AdS/CFT CFT, T>0 What does BH entropy mean? BH Microstates Horizon states Entanglement entropy √ √

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

18. Generalization Field theory BH spacetime R. Brustein, M. Einhorn and A.Y. (to appear)

19. Generalization Field theory BH spacetime f(r 0 )=0   1  ’ 1  ’’ 1    Exp[-S E ] D   (x,0 + ) =  ’ 1 (x)  (x,0 - ) =  ’’ 1 (x)  ’| e -  H |  ’’ 

20. Generalization BH spacetime Field theory ? /2

21. Generalization BH spacetime Field theory Field theory  Field theory /2

22. Summary BH entropy is a result of: –Entanglement –Microstates Counting of states using dual FT’s is consistent with entanglement entropy.

23. End

24. Entanglement entropy S 1 =S 2 Srednicki (1993)

25. AdS/CFT (example) Witten (1998) Massless scalar field in AdS An operator O in a CFT Exp ( )