Black Hole Entropy

• Black holes – a review

• Black holes in string theory

• Entanglement and black holes

• Entanglement in string theory

Amos Yarom

Black holes

A wrong derivation yielding correct results:

R

GM2vescape

If nothing can escape then:

cescapev

Yielding:

12

2

Rc

GMRs=2GM/c2R≤

Black hole condition

x

y

The event horizon

Schwartzshield radius

x

y

t

Event Horizon formed

Schwartzshield radius

Singularityformed

Singularityformed

Event Horizon formed

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

S A

TH=1/(8M)

S = ¼ A

S =0

Microscopic states

What is entropy?

S=k·ln(N)

S=-k r(ln

S=k ln 3S=-k 3 1/3 ln 1/3 = k ln 3

|☺☺O>

|☺O☺>

| O ☺☺>

3

100

03

10

003

1

Macroscopic state

What does black hole entropy mean?

p

x

xp

/ln phaseVS

p

x

?

?

?

String theory

x

y

t

X

X(,)X(,)X(,)

()

String theory

Photon

Massive particle

Graviton

D-branes

Dualities

Dualities

DualitiesSSBH =

An explicit example: AdS/CFT

SCFTNL 4

Ng YMs /4

AdS space CFT

Minkowski space

deSitterAnti deSitter

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)

AdS/CFT

CFT, T>0Anti deSitter +BH

?

Eternal black holes

x

y

t

x

y

t

Event Horizon formed

Schwartzshield radius

Singularityformed

Eternal black holes

Eternal Black holes

t

xt=0

r=0

Eternal Black holes

t

xt=0

r=0

Entanglement entropy

21212

10,0

ie2

1

2/10

02/11

1 2

Results:50% ↑50% ↓

Results ≠0:50% ↑50% ↓

1 2

ie2

1

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)=ln2All |↓22↓| elements

1 2

The vacuum state

|0

t

x

0021 Tr

111 lnS Tr

HeN

1

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. (2005)

LL

212

iii

EEEe

i

|0

iii

E EEe i

11

0021 Trace

Summary

• BH’s have entropy.

• String theory may evaluate this entropy explicitly.

• This evaluation is consistent with entanglement entropy.