Post on 11-Jan-2016
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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!)