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

What is entanglement entropy?

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

21212

10,0

0000

02/12/10

02/12/10

0000

0,00,0

21 Trace

2/10

02/1

S=0

S=Trace (ln1)=ln2S=Trace (ln2)=ln2

All |↓22↓| elements

1 2

2

Black holes

aSinhrgt

aCoshrgx

/)(

/)(

)('

2

)(

0

12

2

rfa

eCarg

r

drfa

r

drfa

feCrh

12

1)(

22122 )()()( drqdrrfdtrfds

f(r0)=0 Coordinate singularity

r0

2222 )())(( drqdxdtrhds

f(0)=- Space-time singularity

“Kruskal” extension

t

x

r=r0

r=0

aSinhrgt

aCoshrgx

/)(

/)(

“Kruskal” extension

aSinhrgt

aCoshrgx

/)(

/)(

22122 )()()( drqdrrfdtrfds

t

x

r=r0

r=0

x

2222 )())(( drqdxdtrhds

The vacuum state

|0

t

x

r=0

r=r0

00inout Tr

outoutout Tr lnS ininin Tr lnS

Finding out

''00')'','(

DLdtExp ][00

(x,0)=(x)

00

x

t

’(x)’’(x)

Trin (’’’out(’1,’’1) =

out’1’’1 Exp[-SE] D

(x,0+) = ’1(x)(x,0-) = ’’1(x)

(x,0+) = ’1(x)2(x)(x,0-) = ’’1(x)2(x)

Exp[-SE] DD2

DLdtExp ][)'','(

(x,0+)=’(x)

(x,0-)=’’(x)

DLdtExp ][)'','(

(x,0+)=’(x)

(x,0-)=’’(x)

Kabat & Strassler (1994), R. Brustein, M. Einhorn and A.Y. (2005)

x

t

’1(x)

’’1(x)

’| e-H|’’

Kabat & Strassler (1994), R. Brustein, M. Einhorn and A.Y. (2005)

Finding in

out’1’’1 Exp[-SE] D

(x,0+) = ’1(x)(x,0-) = ’’1(x)

f ’(r0)

aSinhrgt

aCoshrgx

/)(

/)(

BTZ BH222122 )()()( dJdtrdrrfdrfds

)sinh()(

)cosh()(

argt

argx

22222 )())(( dJdtrdxdtrhds

BTZ BH

)('')(' xex H DSxx Ein ][Exp)('')('

)('')0,(

)(')0,(

xx

xx

),(''),('),(''),(' txttxtxetx tH ),(''),('),(''),(' txttxxtxetx tH

costx tg

gx

i

00

0

DSExptxetx E

xdg

gH d

i

i

][),(''),('

100

0

outi

iiJE nne i i

222122 )()()( dJdtrdrrfdrfds

)sinh()(

)cosh()(

argt

argx

22222 )())(( dJdtrdxdtrhds

)('')(' xex H DSxx Ein ][Exp)('')('

)('')0,(

)(')0,(

xx

xx

t

x

Black hole entanglement entropy

)1(2

1)1(2/)1(

ddVTd

Sd

dd

)ln( outoutTrS

Hout e

22122 )()()( drqdrrfdtrfds

212222 )()()( drfrqdrrfdtds optdV 1

dd

AV

)1( d

ACS

)1(

d

ANCS

4

)(' 0rfT

S.P. de Alwis, N. Ohta, (1995)

What is entanglement entropy?What is entanglement entropy of BH’sHow does string theory evaluate BH entropy?How are these two methods relate to each other?

How to relate them?

NBH G

AS

4

)1( dBH

ANS

?

BH entropy in string theory

SBH SFT(TBH)

=

LS

TBH TFT

=

YMR 4

SBH=A/4

SCFTNL 4

S=A/3

Semiclassical gravity:R>>ls

Free theory: 0

S/A

1/R

AdS BH EntropyS. S. Gubser, I. R. Klebanov, and A. W. Peet (1996)

Anti deSitter +BH

AdS/CFT

CFT, T>0What is entanglement entropy?What is entanglement entropy of BH’sHow does string theory evaluate BH entropy?How are these two methods relate to each other?

How to relate them?

NBH G

AS

4

)1(. dentBH

ANS

NThermal G

AS

4 ?

Thermofield doublesTakahashi and Umezawa, (1975)

) Tr O(O He

i

iiE EEe i

iii

E EOEeO i

HHH IOO

ji

jijiE EIEEOEeO i

,

~~

O

iii

EEEe

i ~2

HTr

.entthermal SS

How to relate them?

NThermal G

AS

4

iii

EEEe

i ~2

NBH G

AS

4

)1(. dentBH

ANS

Nent G

AS

4. ?

Dualities

LS

wedgeHwedgeFT ,H

globalH

wedgeFTwedgeFT ,, HH

LSglobal

wedgewedge HH

globalFT ,H

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

Dualities

globalFT ,H

wedgewedge HH globalH

wedgeFTwedgeFT ,, HH

LSglobal

0 i

jiE EEe i

Tracing

iiE

wedge EEe i

D0

ijiijA

i

ii aa 00Tracing

i

DiDDiDaa 00

i

iDDiE

wedge EEe i

i

DjDiE EEe i

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

Dualities

globalFT ,HglobalH LSglobal

0 i

iiE EEe i

D0

iDiDi

E EEe i

entBHS , entFTS ,=

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

General picture

NThermal G

AS

4

iii

EEEe

i ~2

NBH G

AS

4

)1(. dentBH

ANS

Nent G

AS

4.

Explicit construction: BTZ BH

PeP n2~

12~ J

02

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

B0

BTZ0

03~ J

),,( ),,( tutu

2

tan~)1,,(t

tu *2/2/ ),(),(~ tuetueW LR *2/2/~ RR bebea nne

B

~0 nne

BTZ

2/~0 )1,,( )1,,( tutu

t

Example: AdS BH

212

iii

EEEe

i

AdS BH

AdS/CFT

CFTCFT, T=0CFT, T>0

|0

iii

E EEe i

11

0021 Trace

Example: AdS BH’s

3. A

NCSent

0

0

s

s

l

g)5(. 4 N

ent G

AS

0)5( NG

4/1ssp gll

5

5

R

N

28

5)5(

sgl

RG

s

N

Consequences

dTT

CS V

AS

2ECV

Area scaling

22 ETrE

2 E

R. Brustein and A.Y. (2003)

Area scaling of correlation functions

E E = V V E(x) E(y) ddx ddy

= V V FE(|x-y|) ddx ddy

= D() FE() d

D()= V V (xy) ddx ddy

Geometric term:

Operator dependent term

= D() 2g() d

= - ∂(D()/d-1) d-1 ∂g() d

Geometric term

D()= (r) ddr ddR

R

r ddR V + A2)

(r) ddr d-1 +O(d)

D()=C1Vd-1 ± C2 Ad + O(d+1)

D()= V V (xy) ddx ddy

Area scaling of correlation functions

∂ (D()/d-1)

UV cuttoff at ~1/

D()=C1Vd-1 + C2 Ad + O(d+1)

A

E E = V V E(x) E(y) ddx ddy

= V1 V2 FE(|x-y|) ddx ddy

= D() FE() d

= D() 2g() d

= - ∂(D()/d-1) d-1 ∂g() d

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

Non unitary evolution

21212

10,0

0101

1010

0101

1010

xz SSH

],[ H

))2sin(1(2/10

0))2sin(1(2/1)()( 21 t

ttTrt

dc

baH1 ],[ 111 H ],[ 111 H

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

22 )(~ H22 )(~ outH

nHb ~

],[ GG H

],[ outoutout H

Summary

• BH entropy is a result of:– Entanglement– Microstates

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

End

Entanglement entropy

121

0 aA a

2

)()( 21kk TrTr

S1=S2

Srednicki (1993)

00

,,,, ba

ba AbaA

ba

ba AbaA,,

*TAA

c

cc 00

,,,, ba

ba cAbaAc

,,b

bb AA

†AA

002Tr 001Tr