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Entanglement interpretation of black hole entropy in string theory. Amos Yarom. Ram Brustein. Martin Einhorn. What is entanglement entropy?. What does BH entropy mean?. BH Microstates Entanglement entropy Horizon states. How does it relate to BH entropy?.

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Entanglement interpretation of black hole entropy in string theory

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## Entanglement interpretation of black hole entropy in string theory

Amos Yarom.

Ram Brustein.

Martin Einhorn.

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?

All |↓22↓|

elements

1

2

### Entanglement entropy

S=0

S1=Trace (r1lnr1)=ln2

S2=Trace (r2lnr2)=ln2

r0

### Black holes

f(r0)=0

Coordinate singularity

Space-time singularity

f(0)=-

r=0

t

r=r0

t

x

“Kruskal” extension

r=0

t

r=r0

x

x

### “Kruskal” extension

The vacuum state

r=0

t

r=r0

x

|0

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)

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

### BTZ BH

t

x

BTZ BH

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)

?

### BH entropy in string theory

TBH

TFT

=

SBH

=

SFT(TBH)

Anti deSitter

+BH

CFT

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?

S/A

1/R

Free theory:

l 0

Semiclassical gravity:

R>>ls

S. S. Gubser, I. R. Klebanov, and A. W. Peet (1996)

, T>0

S=A/3

SBH=A/4

How to relate them?

?

?

### Dualities

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

### Dualities

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

Tracing

Tracing

Dualities

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

=

t

q

r

### Explicit construction: BTZ BH

Maldacena and Strominger (1998), Marolf and Louko (1998), Maldacena (2003)

CFTCFT, T=0

CFT, T>0

|0

### Consequences

R. Brustein and A.Y. (2003)

Area scaling

### 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

### 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)

### 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)

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

Srednicki (1993)

S1=S2