On black hole microstates •Introduction •BH entropy •Entanglement entropy •BH microstates Amos Yarom. Ram Brustein. Martin Einhorn.
Jan 11, 2016
On black hole microstates
•Introduction
•BH entropy
•Entanglement entropy
•BH microstates
Amos Yarom.
Ram Brustein.Martin Einhorn.
Geometry
BA
BA
cos A
B
,
BAgBA i
jiij BAgBA
i
iiBABA
General relativity
22
2
sin000
000
00)/21(
10
000/21
r
rrM
rM
g
G=T =0
r=2M
r=0
Coordinate singularity
Spacetime singularity
1000
0100
0010
0001
g
Coordinate singularities
x
y r
10
01g
20
01
rg
x=r cos
y=r sin
Kruskal extension
MSinheMG
Mrt
MCosheMG
Mrx
Mr
Mr
4/2
)2(
4/2
)2(
2/
2/
22
2
sin000
000
000
000
r
r
h
h
g
Mrer
GMh 2/
332
MreMG
Mrxt 2/22
2
)2(
Previous coordinates:
rM2
t
x
r=2M
r=0
t=0
t=1/2
t=1
t=3/2
x
Kruskal extension
t
x
r=2M
r=0
MSinheMG
Mrt
MCosheMG
Mrx
Mr
Mr
4/2
)2(
4/2
)2(
2/
2/
Black hole thermodynamicsJ. Beckenstein (1973) S. Hawking (1975)
S A
TH=1/(8M)
S = ¼ A
S =0
What does BH entropy mean?
• BH Microstates
• Horizon states
• Entanglement entropy
Entanglement entropy
21212
10,0
ie2
1
2/10
02/11
1 2
Results:50% ↑50% ↓
Results ≠0:50% ↑50% ↓
2
1 2
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
The vacuum state
|0
t
x
r=0
r=2M
0021 Tr
111 lnS Tr 222 lnS Tr
Finding 1
''00')'','(
DLdtExp ][00
(x,0)=(x)
00
x
t
’(x)’’(x)
Tr2 (’’’1(’1,’’1) =
1’1’’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)
What does BH entropy mean?• BH Microstates
• Horizon states
• Entanglement entropy
√x
t
’1(x)
’’1(x)
’| e-H|’’
Kabbat & Strassler (1994), R. Brustein, M. Einhorn and A.Y. (to appear)
Finding 1
1’1’’1 Exp[-SE] D
(x,0+) = ’1(x)(x,0-) = ’’1(x)
MSinheMG
Mrt
MCosheMG
Mrx
Mr
Mr
4/2
)2(
4/2
)2(
2/
2/
Counting of microstates
(Conformal) field theoryCurved spacetime
Quantized gravity
4 L
String theory
AdS/CFT
SCFTNL 4
Ng YMs /4
AdS space CFT
Minkowski space
deSitterAnti deSitter
O
Z(b=0) Exp(OdV)=
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)
Anti deSitter +BH
AdS/CFT
CFT, T>0
What does BH entropy mean?• BH Microstates
• Horizon states
• Entanglement entropy
√
√
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. (to appear)
Generalization
)(00
0)(/10
00)(
rq
rf
rf
g
aSinhrgt
aCoshrgx
/)(
/)(
)('
2
)(
0
12
2
rfa
eCarg
r
drfa
Field theory
L
BH spacetime
f(r0)=0
)(00
0)(0
00)(
rq
rh
rh
g 22
12
1
)(
)(
txrg
feCrh
r
drfa
1’1’’1 Exp[-SE] D
(x,0+) = ’1(x)(x,0-) = ’’1(x)
’| e-H|’’
GeneralizationBH spacetime
HeTr 100
BH spacetime Field theory
L?
dHdd eTr 100
LΗ d
/2
2120 ii
i
E
d EEei
GeneralizationBH spacetime
HeTr 100
BH spacetime
Field theory
dHdd eTr 100
Field theory Field theory
LLH d
/2
2120 ii
i
E
d EEei
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
AdS/CFT (example)
dVOExpZ b 00 )( )(
0 )( Ib eZ
xdDDgI d 1
2
1)(
Witten (1998)
Massless scalar field in AdS An operator O in a CFT
0
DD
')'('
),( 0220
00 xdx
xxx
xcxx d
d
d
xx
xxcdI 2
00
'
)'()(
2)(
')'()',()(
2
1 00
0
xxddxxxGxExp
dVOExp
dd
)'()()',( xOxOxxG
dxx
cdxOxO 2
'
1
2)'()(
dVOExp 0
d
dd
xx
xxcdxxddxxxGx 2
0000
'
)'()(
2')'()',()(
2
1
d
xx
xxcd2
00
'
)'()(
2
Exp( )