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

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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)


?

How to relate them?


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?

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


How to relate them?

?


Thermofield doublesTakahashi and Umezawa, (1975)


?

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)

=


General picture


t

q

r

Explicit construction: BTZ BH

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


AdS/CFT

Example: AdS BH

AdS BH

CFTCFT, T=0

CFT, T>0

|0


Example: AdS BH’s


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


Entanglement entropy

Srednicki (1993)

S1=S2


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