Entanglement interpretation of black hole entropy in string theory

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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

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Entanglement interpretation of black hole entropy in string theory

Amos Yarom.

Ram Brustein.

Martin Einhorn.

What is entanglement entropy?

- 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

S=0

S1=Trace (r1lnr1)=ln2

S2=Trace (r2lnr2)=ln2

r0

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

The vacuum state

r=0

t

r=r0

x

|0

Trin(y’ y’’

rout(y’1,y’’1) =

Exp[-SE] DfD2

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

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)

?

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?

AdS/CFT

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?

?

?

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

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

Tracing

Tracing

Dualities

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

=

t

q

r

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

AdS/CFT

AdS BH

CFTCFT, T=0

CFT, T>0

|0

R. Brustein and A.Y. (2003)

Area scaling

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(xxy) ddx ddy

D(x)= V V d(xxy) ddx ddy

D(x)= d(xr) ddr ddR

ddR V + Ax +O(x2)

d(xr) ddr xd-1 +O(xd)

D(x)=C1Vxd-1 ± C2 Axd + O(xd+1)

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)

- 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