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Entanglement interpretation of black hole entropy in string
theory
Amos Yarom.
Ram Brustein.Martin Einhorn.
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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?
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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
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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
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“Kruskal” extension
t
x
r=r0
r=0
aSinhrgt
aCoshrgx
/)(
/)(
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“Kruskal” extension
aSinhrgt
aCoshrgx
/)(
/)(
22122 )()()( drqdrrfdtrfds
t
x
r=r0
r=0
x
2222 )())(( drqdxdtrhds
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The vacuum state
|0
t
x
r=0
r=r0
00inout Tr
outoutout Tr lnS ininin Tr lnS
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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)
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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
/)(
/)(
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BTZ BH222122 )()()( dJdtrdrrfdrfds
)sinh()(
)cosh()(
argt
argx
22222 )())(( dJdtrdxdtrhds
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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
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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?
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How to relate them?
NBH G
AS
4
)1( dBH
ANS
?
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BH entropy in string theory
SBH SFT(TBH)
=
LS
TBH TFT
=
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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?
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How to relate them?
NBH G
AS
4
)1(. dentBH
ANS
NThermal G
AS
4 ?
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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
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How to relate them?
NThermal G
AS
4
iii
EEEe
i ~2
NBH G
AS
4
)1(. dentBH
ANS
Nent G
AS
4. ?
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Dualities
LS
wedgeHwedgeFT ,H
globalH
wedgeFTwedgeFT ,, HH
LSglobal
wedgewedge HH
globalFT ,H
R. Brustein, M. Einhorn and A.Y. (2005)
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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)
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Dualities
globalFT ,HglobalH LSglobal
0 i
iiE EEe i
D0
iDiDi
E EEe i
entBHS , entFTS ,=
R. Brustein, M. Einhorn and A.Y. (2005)
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General picture
NThermal G
AS
4
iii
EEEe
i ~2
NBH G
AS
4
)1(. dentBH
ANS
Nent G
AS
4.
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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
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Example: AdS BH
212
iii
EEEe
i
AdS BH
AdS/CFT
CFTCFT, T=0CFT, T>0
|0
iii
E EEe i
11
0021 Trace
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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
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Consequences
dTT
CS V
AS
2ECV
Area scaling
22 ETrE
2 E
R. Brustein and A.Y. (2003)
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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
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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
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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
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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
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ConsequencesR. Brustein M. Einhorn and A.Y. (in progress)
22 )(~ H22 )(~ outH
nHb ~
],[ GG H
],[ outoutout H
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Summary
• BH entropy is a result of:– Entanglement– Microstates
• Counting of states using dual FT’s is consistent with entanglement entropy.
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End
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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
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