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

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

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

Black holes f(r 0 )=0Coordinate singularity r0r0 f(0)=-  Space-time singularity

“Kruskal” extension t x r=r 0 r=0 

“Kruskal” extension t x r=r 0 r=0 x

The vacuum state |0  t x r=0 r=r 0

Finding  out  (x,0)=  (x) x t  ’(x)  ’’(x) Tr in  (  ’  ’’   out (  ’ 1,  ’’ 1 ) =  out  ’ 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) Kabat & Strassler (1994), R. Brustein, M. Einhorn and A.Y. (2005)

x t  ’ 1 (x)  ’’ 1 (x)  ’| e -  H |  ’’  Kabat & Strassler (1994), R. Brustein, M. Einhorn and A.Y. (2005) Finding  in  out  ’ 1  ’’ 1    Exp[-S E ] D   (x,0 + ) =  ’ 1 (x)  (x,0 - ) =  ’’ 1 (x)    f ’(r 0 ) 

BTZ BH

t x

Black hole entanglement entropy S.P. de Alwis, N. Ohta, (1995) 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?

How to relate them? ?

BH entropy in string theory S BH S FT (T BH ) = T BH T FT =

S BH =A/4 S=A/3 Semiclassical gravity: R>> l s 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 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?

How to relate them? ?

Thermofield doubles Takahashi and Umezawa, (1975)

How to relate them? ?

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

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

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

General picture

Explicit construction: BTZ BH Maldacena and Strominger (1998), Marolf and Louko (1998), Maldacena (2003) t  

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

Example: AdS BH’s

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

Area scaling of correlation functions  E E  =  V  V  E(x) E(y)  d d x d d y =  V  V F E (|x-y|) d d x d d y =  D(  ) F E (  ) d  D(  )=  V  V  (  x  y  ) d d x d d y Geometric term: Operator dependent term =  D(  )  2 g(  ) d  = -  ∂  (D(  )/  d-1 )  d-1 ∂  g(  ) d 

Geometric term D(  )=    (  r) d d r d d R  d d R  V + A  2 )  (  r) d d r   d-1 +O(  d ) D(  )=C 1 V  d-1 ± C 2 A  d + O(  d+1 ) D(  )=  V  V  (  x  y  ) d d x d d y

Area scaling of correlation functions  ∂  (D(  )/  d-1 )   UV cuttoff at  ~1/  D(  )=C 1 V  d-1 + C 2 A  d + O(  d+1 )  A  E E  =  V  V  E(x) E(y)  d d x d d y =  V1  V2 F E (|x-y|) d d x d d y =  D(  ) F E (  ) d  =  D(  )  2 g(  ) d  = -  ∂  (D(  )/  d-1 )  d-1 ∂  g(  ) d 

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

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

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

End

Entanglement entropy S 1 =S 2 Srednicki (1993)