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?.

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Entanglement interpretation of black hole entropy in string theory

Entanglement interpretation of black hole entropy in string theory

Amos Yarom.

Ram Brustein.

Martin Einhorn.


What does bh entropy mean

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?


Entanglement entropy

All |↓22↓|

elements

1

2

Entanglement entropy

S=0

S1=Trace (r1lnr1)=ln2

S2=Trace (r2lnr2)=ln2


Black holes

r0

Black holes

f(r0)=0

Coordinate singularity

Space-time singularity

f(0)=-


Entanglement interpretation of black hole entropy in string theory

r=0

t

r=r0

t

x

“Kruskal” extension


Kruskal extension

r=0

t

r=r0

x

x

“Kruskal” extension


Entanglement interpretation of black hole entropy in string theory

The vacuum state

r=0

t

r=r0

x

|0


Entanglement interpretation of black hole entropy in string theory

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)


Entanglement interpretation of black hole entropy in string theory

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

BTZ BH


Entanglement interpretation of black hole entropy in string theory

t

x

BTZ BH


Entanglement interpretation of black hole entropy in string theory

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)


How to relate them

?

How to relate them?


Bh entropy in string theory

BH entropy in string theory

TBH

TFT

=

SBH

=

SFT(TBH)


Ads bh entropy

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


Entanglement interpretation of black hole entropy in string theory

How to relate them?

?


Thermofield doubles takahashi and umezawa 1975

Thermofield doublesTakahashi and Umezawa, (1975)


How to relate them1

?

How to relate them?


Dualities

Dualities

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


Dualities1

Dualities

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

Tracing

Tracing


Entanglement interpretation of black hole entropy in string theory

Dualities

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

=


General picture

General picture


Explicit construction btz bh

t

q

r

Explicit construction: BTZ BH

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


Example ads bh

AdS/CFT

Example: AdS BH

AdS BH

CFTCFT, T=0

CFT, T>0

|0


Example ads bh s

Example: AdS BH’s


Consequences

Consequences

R. Brustein and A.Y. (2003)

Area scaling


Area scaling of correlation functions

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

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 functions1

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)


Entanglement interpretation of black hole entropy in string theory

Consequences

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

Non unitary evolution


Entanglement interpretation of black hole entropy in string theory

Consequences

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


Summary

Summary

  • BH entropy is a result of:

    • Entanglement

    • Microstates

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


Entanglement interpretation of black hole entropy in string theory

End


Entanglement entropy1

Entanglement entropy

Srednicki (1993)

S1=S2


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