Wald’s Entropy, Area & Entanglement Introduction: –Wald’s Entropy –Entanglement entropy in space-time Wald’s entropy is (sometimes) an area ( of some metric)

Wald’s Entropy, Area & Entanglement

• Introduction: – Wald’s Entropy– Entanglement entropy in space-time

• Wald’s entropy is (sometimes) an area (of some metric) or related to the area by a multiplicative factor

• Relating Wald’s entropy to Entanglement entropy

אוניברסיטת בן-גוריון

Ram BrusteinR.B., MERAV HADAD

=====================

R.B, Einhorn, Yarom,

0508217, 0609075

Series of papers with Yarom, (also David Oaknin)

• What is Wald’s entropy ?

• How to evaluate Wald’s entropy– The Noether charge Method (W ‘93, LivRev

2001+…) – The field redefinition method (JKM, ‘93)

• What is entanglement entropy ?– How is it related to BH entropy ?– How to evaluate entanglement entropy ?

• How are the two entropies related ?

Plan

Result: for a class of theories both depend on the geometry in the same way, and can be made equal by a choice of scale

Wald’s entropy

• – Bifurcating Killing Horizon:

d-1 space-like surface @ intersection of two KH’s (d = D-1=# of space dimensions)– Killing vector vanishes on the surface

• The binormal vector ab : normal to the tangent & normal of

• Functional derivative as if Rabcd and gab are independent

•

1 1Wald 2d d

ab cdabcd

LS hd x

R

det , ,ijh g i j surface

1 , 2abab a b abD

Wald’s entropy

Properties:

•Satisfies the first law

•Linear in the “correction terms”

•Seems to agree with string theory counting

Wald2M S J

1( , )

16L R F R

G

1 1Wald 2d d

ab cdabcd

LS hd x

R

1L

16R g

G

2 2 1 2 1( ) ( ) ( ) dd s f r d f r dr q r d x

.

( ) ( ) ( )s sf r r r f r

( ) 0sf r

Wald’s entropy: the simplest example

The bifurcation surface t =0, r = rs

1 1

1 1

cab a b bc a

ba b a a b ab

D g D

D D

1 0 0 001 10 00 10

11 11

1 2 221 21 22 12

1 3 331 13 33 13

(1,0,0,0)

( )2( ) 1

( ) 2 ( )

0

0

0

a

s

s s

f rg f r

f r f r

g

g

116L G R g

R g g R

.

The simplest example:

Wald

1

2

111 00 00 11 2

01 01 10 10

1

2

2

2( )

16

1( )

8

1( )

4 4

ab cdabcd

d

s

d

s

d

s

LS h

R

g g q rG

g g g g q rG

Aq r

G G

2 2 1 2 1( ) ( ) ( ) dd s f r d f r dr q r d x

1L ,

16 aR g F R D R gG

.

( ) ( ) ( )s sf r r r f r ( ) 0sf r

A more complicate example

;24

abcd e abcdWald e ab cd

AS Y D Z h

G

,

abcd ab cd

abcd abcd

F F R FY g g

R R R R

;e abcd ab cde

e abcd e e abcd e

D RF F FZ g g

D R D R D R D R

24

44

ab cde ab cd

e

ee

A F FS D g g h

G R D R

A F FD h

G R D R

2 2 1 2 1( ) ( ) ( ) dd s f r d f r dr q r d x

The field redefinition method for evaluating Wald’s entropy

• The idea (Jacobson, Kang, Myers, gr-qc/9312023)– Make a field redifinition

– Simplify the action (for example to Einstein’s GR)

• Conditions for validity– The Killing horizons, bifurcation surface, and asymptotic

structure are the same before and after

– Guaranteed when ab is constructed from the original metric and matter fields Lab= 0 and ab vanishes sufficiently rapidly

ab ab ab abg g g

A more2 complicated Example:

For a1=0 Weyl transformation

is the metric in the subspace normal to the horizon

The entanglement interpretation:

• The statistical properties of space-times with causal boundaries arise because classical observers in them have access only to a part of the whole quantum state trace over the classically inaccessible DOF

( “Microstates are due to entanglement” )

• The fundamental physical objects describing the physics of space-times with causal boundaries are their global quantum state and the unitary evolution operator.

( “Entropy is in the eyes of the beholder” )

The entanglement interpretation:• Properties:

–Observer dependent–Area scaling–UV sensitive–Depends on the matter content, # of fields …,

Entanglement

21212

10,0

0000

02/12/10

02/12/10

0000

0,00,0

21 Trace

2/10

02/1

S=0

S1=-Tr (1ln1)=ln2

S2=-Trace (2ln2)=ln2

All |↓22↓| elements

1 2

2

ininin OTrO

inout

outin

Tr

Tr

IOO

OIO

in

out

Entanglement

iii

E

outin EEeZ

HΗi

21

&

outin HHΗ

If : thermal & time translation invariance then TFD:

purification

r = r s

= 0

= const.

r = const.

Entanglement in space-time

Examples: Minkowski, de Sitter, Schwarzschild, non-rotating BTZ BH, can be extended to rotating, charged, non-extremal BHs

“Kruskal” extension

“Kruskal” extension

aSinhrgt

aCoshrgx

/)(

/)(

2 2 1 2 2( ) ( ) ( )ds f r d f r dr q r d

t

x

r = rs

r = 0

x

2222 )())(( drqdxdtrhds

The vacuum state

|0

t

x

r=0

r = rs

00inout Tr

outoutout Tr lnS ininin Tr lnS

r = rs

= 0

= const.

r = const.

r = rs

= 0

= const.

r = const.

Two ways of calculating in

Kabat & Strassler (flat space) Jacobson

Construct the HH vacuum: the invariant regular state

inoutinout

R.B., M. Einhorn and A.Yarom

1. The boundary conditions are the same2. The actions are equal3. The measures are equal

effHein

0

Results*:

If

Then

Heff – generator of (Imt) time translations

* Method works for more general cases

' '' 1, 0 ( , )

( , ,0), 0 ( , )

exps in

s in

dr r r xin in in r xr r r x

D d drdx L

0

0

0

1/ 4' '' 1( ,0) ( , )

( , ) ( , ) 0

expeff

in

in

H dx x xin in

x x x

e D g d d dx L

Sis divergent Naïve origin: divergence of the optical volume near the horizon, *not* brick wall.Choice of S=A/4G

Entanglement entropy

ininin TrS ln

– proper length short distance cutoff in optical metric

Emparande Alwis & Ohta

EXPLAIN !!!!

Extensions, Consequences

1. Works for Eternal AdS BH’s, consistent with AdS-CFT, RB, Einhorn, Yarom

2. Rotating and charged BHs, RB, Einhorn, Yarom 3. Extremal BHs (on FT side): Marolf and Yarom

4. Non-unitary evolution : RB, Einhorn, Yarom

Relating Wald’s entropy to Entanglement entropy

• Wald’s entropy is an area for some metric or related to the area by a multiplicative factor– So far: have been able to show this for theories that can

be brought to Einstein’s by a metric redefinition equivalent to a conformal rescaling in the r-t plane on the horizon.

• Entanglement entropy scales as the area • Changes in the minimal length account for the

differences

Relating Wald’s entropy to Entanglement entropy

• Example : more complicated matter action– Changes in the matter action do not change

Wald’s entropy– Changes in the matter action do not change the

entanglement entropy (as long as the matter kinetic terms start with a canonical term).

• Example : theories that can be related by a Weyl transformation to Einstein + (conformal) matter

2g g

, , ,...D nabcd abcdL d x g F R D R DL d x g R

2 3 4

; ; ;

, , ,...

2 1 1 4

D nabcd abcd

D D

L d x g F R D R

d x g R D g D D g

, , ,...nabcd abcdR D R

1Wald 2 d

ab cdabcd

FS hd x

R

2 1 2... ; ; ;

2 1 2... ; ; ;

2 1 2... ; ; ;

2 2 1 2... ; ; ;

2 1 1 4

2 1 1 4

2 1 1 4

1 2 4

DR

abcd

DR

DR

D ac bd DR

FR D g D D g

R

R D g D D g

R D g D D g

g g D D

g

2 3... ; ... ;

2 1 2... ; ; ;

2 1 ln

2 1 1 4

D ac bd DR R

abcd

DR

Fg g D D g

R

R D g D D g

2 3... ; ... ;

2 1 2... ; ; ;

2 1 ln

2 1 1 4

D ac bd DR R

abcd

DR

Fg g D D g

R

R D g D D g

1Wald

2 3... ; ... ; 1

2 1 2... ; ; ;

2 1

3...

2

2 1 ln2

2 1 1 4

2

2 2 1

dab cd

abcd

D ac bd DR R d

ab cdDR

D ac bd dab cd

DR

FS hd x

R

g g D D ghd x

R D g D D g

g g hd x

D D

; ... ;lnR g

1

2 1 2 1... ; ; ;

2 1 2 1...

2 2 1 1 4

2 2

dab cd

D dR ab cd

D ac bd d D dab cd R ab cd

hd x

R D g D D g hd x

g g hd x R hd x

0R

1 1Wald

1 1 1

2

4 4 4

d ac bd dab cd

d d d

S g g hd x

hd x hd x A

1Wald

2 1 2 1...

2

2 2

dab cd

abcd

D ac bd d D dab cd R ab cd

FS hd x

R

g g hd x R hd x

Relating Wald’s entropy to Entanglement entropy

• Example : theories that can be related by a Weyl transformation to Einstein + (conformal) matter

2( , )

16 16D R

L d x g F RG G

Wald 1 ( , )4 4 R

A AS F R

G G

Entanglement 2D

AS

By a consistent choice of make

Entanglement 4

AS

G

1 ( , )g g f R

2( , )

16D R

L d x g F RG

2( , )

16D R

L d x g y TG

1 ( 2)4

D

G

1 ( , )|

f Rrs

Entanglement

Wald

21 ( , )

|4 2

1 ( , )|4 R

A DS f R

rG s

AF R S

rG s

Entanglement Wald4

AS S

G

JKM: It is always possible to find (to first order in ) a function2

( , ) ( , )2 R

Df R F R

Relating Wald’s entropy to Entanglement entropy

• Example:– More complicated– The transformation is not conformal– The transformation is only conformal on r-t part

of the metric, and only on the horizon– Works in a similar way to the fully conformal

transformation

2

16D ab

ab

RL d x g R R

G

Summary

1. Wald’s entropy is consistent with entanglement entropy

2. Wald’s entropy is (sometimes) an area (for some metric) or related to the area by a multiplicative factor

3. BH Entropy can be interpreted as entanglement entropy (not a correction!)