entanglement interpretation of black hole entropy in string theory amos yarom. ram brustein. martin...
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Entanglement interpretation of black hole entropy in string
theory
Amos Yarom.
Ram Brustein.Martin Einhorn.
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?
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
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
“Kruskal” extension
t
x
r=r0
r=0
aSinhrgt
aCoshrgx
/)(
/)(
“Kruskal” extension
aSinhrgt
aCoshrgx
/)(
/)(
22122 )()()( drqdrrfdtrfds
t
x
r=r0
r=0
x
2222 )())(( drqdxdtrhds
The vacuum state
|0
t
x
r=0
r=r0
00inout Tr
outoutout Tr lnS ininin Tr lnS
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)
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
/)(
/)(
BTZ BH222122 )()()( dJdtrdrrfdrfds
)sinh()(
)cosh()(
argt
argx
22222 )())(( dJdtrdxdtrhds
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
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?
How to relate them?
NBH G
AS
4
)1( dBH
ANS
?
BH entropy in string theory
SBH SFT(TBH)
=
LS
TBH TFT
=
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?
How to relate them?
NBH G
AS
4
)1(. dentBH
ANS
NThermal G
AS
4 ?
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
How to relate them?
NThermal G
AS
4
iii
EEEe
i ~2
NBH G
AS
4
)1(. dentBH
ANS
Nent G
AS
4. ?
Dualities
LS
wedgeHwedgeFT ,H
globalH
wedgeFTwedgeFT ,, HH
LSglobal
wedgewedge HH
globalFT ,H
R. Brustein, M. Einhorn and A.Y. (2005)
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)
Dualities
globalFT ,HglobalH LSglobal
0 i
iiE EEe i
D0
iDiDi
E EEe i
entBHS , entFTS ,=
R. Brustein, M. Einhorn and A.Y. (2005)
General picture
NThermal G
AS
4
iii
EEEe
i ~2
NBH G
AS
4
)1(. dentBH
ANS
Nent G
AS
4.
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
Example: AdS BH
212
iii
EEEe
i
AdS BH
AdS/CFT
CFTCFT, T=0CFT, T>0
|0
iii
E EEe i
11
0021 Trace
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
Consequences
dTT
CS V
AS
2ECV
Area scaling
22 ETrE
2 E
R. Brustein and A.Y. (2003)
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
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
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
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
ConsequencesR. Brustein M. Einhorn and A.Y. (in progress)
22 )(~ H22 )(~ outH
nHb ~
],[ GG H
],[ outoutout H
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