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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? theory

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 |↓ theory22↓|

elements

1

2

Entanglement entropy

S=0

S1=Trace (r1lnr1)=ln2

S2=Trace (r2lnr2)=ln2


Black holes

r theory0

Black holes

f(r0)=0

Coordinate singularity

Space-time singularity

f(0)=-


r=0 theory

t

r=r0

t

x

“Kruskal” extension


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
BTZ BH theory


t theory

x

BTZ BH


What is entanglement entropy? theory

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

? theory

How to relate them?


Bh entropy in string theory
BH entropy in string theory theory

TBH

TFT

=

SBH

=

SFT(TBH)


Ads bh entropy

theory

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



Thermofield doubles takahashi and umezawa 1975
Thermofield doubles theoryTakahashi and Umezawa, (1975)


How to relate them1

? theory

How to relate them?


Dualities
Dualities theory

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


Dualities1
Dualities theory

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

Tracing

Tracing


Dualities theory

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

=



Explicit construction btz bh

t theory

q

r

Explicit construction: BTZ BH

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


Example ads bh

AdS/CFT theory

Example: AdS BH

AdS BH

CFTCFT, T=0

CFT, T>0

|0



Consequences
Consequences theory

R. Brustein and A.Y. (2003)

Area scaling


Area scaling of correlation functions
Area scaling of correlation functions theory

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 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)


Area scaling of correlation functions1
Area scaling of correlation functions theory

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
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 entropy1
Entanglement entropy theory

Srednicki (1993)

S1=S2


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