Entanglement interpretation of black hole entropy in string theory

1. Entanglement interpretation of black hole entropy in string theory Amos Yarom. Ram Brustein. Martin Einhorn.

2. What is entanglement entropy? What does BH entropy mean? • BH Microstates • Entanglement entropy • Horizon states How does it relate to BH entropy? How does string theory evaluate BH entropy? How are these two methods relate to each other?

3. All |↓22↓| elements 1 2 Entanglement entropy S=0 S1=Trace (r1lnr1)=ln2 S2=Trace (r2lnr2)=ln2

4. r0 Black holes f(r0)=0 Coordinate singularity Space-time singularity f(0)=-

5. r=0 t r=r0 t x “Kruskal” extension

6. r=0 t r=r0 x x “Kruskal” extension

7. The vacuum state r=0 t r=r0 x |0

8. Trin(y’ y’’ rout(y’1,y’’1) =   Exp[-SE] DfD2 f(x,0+)=y’(x) f(x,0)=y(x) f(x,0+)=y’(x) f(x,0-)=y’’(x) t f(x,0-)=y’’(x) out y’1 y’’1 Exp[-SE] Df f(x,0+) = y’1(x)y2(x) y’(x) y’’(x) f(x,0-) = y’’1(x)y2(x) x f(x,0+) = y’1(x) f(x,0-) = y’’1(x) Finding rout Kabat & Strassler (1994), R. Brustein, M. Einhorn and A.Y. (2005)

9. t out y’1 y’’1 Exp[-SE] Df y’1(x) x y’’1(x) f(x,0+) = y’1(x) f(x,0-) = y’’1(x) Finding rin Kabat & Strassler (1994), R. Brustein, M. Einhorn and A.Y. (2005)  ’| e-bH|’’ b=T-1=f ’(r0)/4p

10. BTZ BH

11. t x BTZ BH

12. What is entanglement entropy? What is entanglement entropy of BH’s How does string theory evaluate BH entropy? How are these two methods relate to each other? Black hole entanglement entropy S.P. de Alwis, N. Ohta, (1995)  

13. ? How to relate them?

14. BH entropy in string theory TBH TFT = SBH = SFT(TBH)

15.  Anti deSitter +BH CFT What is entanglement entropy?  What is entanglement entropy of BH’s How does string theory evaluate BH entropy? AdS/CFT How are these two methods relate to each other? S/A 1/R Free theory: l 0 Semiclassical gravity: R>>ls AdS BH Entropy S. S. Gubser, I. R. Klebanov, and A. W. Peet (1996) , T>0  S=A/3 SBH=A/4

16. How to relate them? ?

17. Thermofield doublesTakahashi and Umezawa, (1975)

18. ? How to relate them?

19. Dualities R. Brustein, M. Einhorn and A.Y. (2005)

20. Dualities R. Brustein, M. Einhorn and A.Y. (2005) Tracing Tracing

21. Dualities R. Brustein, M. Einhorn and A.Y. (2005) =

22. General picture

23. t q r Explicit construction: BTZ BH Maldacena and Strominger (1998), Marolf and Louko (1998), Maldacena (2003)

24. AdS/CFT Example: AdS BH AdS BH CFTCFT, T=0 CFT, T>0 |0

25. Example: AdS BH’s

26. Consequences R. Brustein and A.Y. (2003) Area scaling

27. Area scaling of correlation functions EE =  V  V E(x) E(y) ddx ddy = V  V FE(|x-y|) ddx ddy = D(x) FE(x) dx = D(x) 2g(x) dx = - ∂x(D(x)/xd-1) xd-1 ∂xg(x) dx Geometric term: Operator dependent term D(x)=V V d(xxy) ddx ddy

28. Geometric term D(x)= V  V d(xxy) ddx ddy D(x)=  d(xr) ddr ddR ddR  V + Ax +O(x2) d(xr) ddr  xd-1 +O(xd) D(x)=C1Vxd-1 ± C2 Axd + O(xd+1)

29. Area scaling of correlation functions EE =  V  V E(x) E(y) ddx ddy = V1  V2 FE(|x-y|) ddx ddy = D(x) FE(x) dx = D(x) 2g(x) dx = - ∂x(D(x)/xd-1) xd-1 ∂xg(x) dx UV cuttoff at x~1/L  ∂ x(D(x)/xd-1) 1/L   A D(x)=C1Vxd-1 + C2 Axd + O(xd+1)

30. Consequences R. Brustein M. Einhorn and A.Y. (in progress) Non unitary evolution

31. Consequences R. Brustein M. Einhorn and A.Y. (in progress)

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

33. End

34. Entanglement entropy Srednicki (1993) S1=S2