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Entanglement interpretation of black hole entropy in string theory - PowerPoint PPT Presentation

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

Amos Yarom.

Ram Brustein.

Martin Einhorn.

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 |↓ theory22↓|

elements

1

2

Entanglement entropy

S=0

S1=Trace (r1lnr1)=ln2

S2=Trace (r2lnr2)=ln2

r theory0

Black holes

f(r0)=0

Coordinate singularity

Space-time singularity

f(0)=-

r=0 theory

t

r=r0

t

x

“Kruskal” extension

r=0 theory

t

r=r0

x

x

“Kruskal” extension

The vacuum state theory

r=0

t

r=r0

x

|0

Tr theoryin(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 theory

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 theory

t theory

x

BTZ BH

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)

? theory

How to relate them?

TBH

TFT

=

SBH

=

SFT(TBH)

theory

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

Thermofield doubles theoryTakahashi and Umezawa, (1975)

? theory

How to relate them?

Dualities theory

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

Dualities theory

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

Tracing

Tracing

Dualities theory

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

=

t theory

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 theory

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(xxy) ddx ddy

Geometric term theory

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)

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 theory

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

Non unitary evolution

Consequences theory

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

Summary theory

• BH entropy is a result of:

• Entanglement

• Microstates

• Counting of states using dual FT’s is consistent with entanglement entropy.

End theory

Entanglement entropy theory

Srednicki (1993)

S1=S2