On black hole microstates

•Introduction

•BH entropy

•Entanglement entropy

•BH microstates

Amos Yarom.

Ram Brustein.Martin Einhorn.

Geometry

BA

BA

cos A

B

,

BAgBA i

jiij BAgBA

i

iiBABA

General relativity

22

2

sin000

000

00)/21(

10

000/21

r

rrM

rM

g

G=T =0

r=2M

r=0

Coordinate singularity

Spacetime singularity

1000

0100

0010

0001

g

Coordinate singularities

x

y r

10

01g

20

01

rg

x=r cos

y=r sin

Kruskal extension

MSinheMG

Mrt

MCosheMG

Mrx

Mr

Mr

4/2

)2(

4/2

)2(

2/

2/

22

2

sin000

000

000

000

r

r

h

h

g

Mrer

GMh 2/

332

MreMG

Mrxt 2/22

2

)2(

Previous coordinates:

rM2

t

x

r=2M

r=0

t=0

t=1/2

t=1

t=3/2

x

Kruskal extension

t

x

r=2M

r=0

MSinheMG

Mrt

MCosheMG

Mrx

Mr

Mr

4/2

)2(

4/2

)2(

2/

2/

Black hole thermodynamicsJ. Beckenstein (1973) S. Hawking (1975)

S A

TH=1/(8M)

S = ¼ A

S =0

What does BH entropy mean?

• BH Microstates

• Horizon states

• Entanglement entropy

Entanglement entropy

21212

10,0

ie2

1

2/10

02/11

1 2

Results:50% ↑50% ↓

Results ≠0:50% ↑50% ↓

2

1 2

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

The vacuum state

|0

t

x

r=0

r=2M

0021 Tr

111 lnS Tr 222 lnS Tr

Finding 1

''00')'','(

DLdtExp ][00

(x,0)=(x)

00

x

t

’(x)’’(x)

Tr2 (’’’1(’1,’’1) =

1’1’’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)

What does BH entropy mean?• BH Microstates

• Horizon states

• Entanglement entropy

√x

t

’1(x)

’’1(x)

’| e-H|’’

Kabbat & Strassler (1994), R. Brustein, M. Einhorn and A.Y. (to appear)

Finding 1

1’1’’1 Exp[-SE] D

(x,0+) = ’1(x)(x,0-) = ’’1(x)

MSinheMG

Mrt

MCosheMG

Mrx

Mr

Mr

4/2

)2(

4/2

)2(

2/

2/

Counting of microstates

(Conformal) field theoryCurved spacetime

Quantized gravity

4 L

String theory

AdS/CFT

SCFTNL 4

Ng YMs /4

AdS space CFT

Minkowski space

deSitterAnti deSitter

O

Z(b=0) Exp(OdV)=

YMR 4

Maldacena (1997)

YMR 4

SBH=A/4

SCFTNL 4

S=A/3

Semiclassical gravity:R>>’

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>0

What does BH entropy mean?• BH Microstates

• Horizon states

• Entanglement entropy

√

√

AdS BH

212

iii

EEEe

i

SCFTNL 4

AdS BH

AdS/CFT

CFTCFT, T=0CFT, T>0

?

|0

iii

E EEe i

11

0021 Trace

Maldacena (2003)

GeneralizationField theoryBH spacetime

L

R. Brustein, M. Einhorn and A.Y. (to appear)

Generalization

)(00

0)(/10

00)(

rq

rf

rf

g

aSinhrgt

aCoshrgx

/)(

/)(

)('

2

)(

0

12

2

rfa

eCarg

r

drfa

Field theory

L

BH spacetime

f(r0)=0

)(00

0)(0

00)(

rq

rh

rh

g 22

12

1

)(

)(

txrg

feCrh

r

drfa

1’1’’1 Exp[-SE] D

(x,0+) = ’1(x)(x,0-) = ’’1(x)

’| e-H|’’

GeneralizationBH spacetime

HeTr 100

BH spacetime Field theory

L?

dHdd eTr 100

LΗ d

/2

2120 ii

i

E

d EEei

GeneralizationBH spacetime

HeTr 100

BH spacetime

Field theory

dHdd eTr 100

Field theory Field theory

LLH d

/2

2120 ii

i

E

d EEei

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

AdS/CFT (example)

dVOExpZ b 00 )( )(

0 )( Ib eZ

xdDDgI d 1

2

1)(

Witten (1998)

Massless scalar field in AdS An operator O in a CFT

0

DD

')'('

),( 0220

00 xdx

xxx

xcxx d

d

d

xx

xxcdI 2

00

'

)'()(

2)(

')'()',()(

2

1 00

0

xxddxxxGxExp

dVOExp

dd

)'()()',( xOxOxxG

dxx

cdxOxO 2

'

1

2)'()(

dVOExp 0

d

dd

xx

xxcdxxddxxxGx 2

0000

'

)'()(

2')'()',()(

2

1

d

xx

xxcd2

00

'

)'()(

2

Exp( )