Transcript The Kerr/CFT Correspondence

Black Hole Decay in the
Kerr/CFT Correspondence
Tom Hartman
Harvard University
0809.4266 TH, Guica, Song, and Strominger
and work in progress with Song and Strominger
ESI Workshop on Gravity in Three Dimensions
April 2009
Introduction
Near-Extremal Kerr/CFT
Black hole decay
- CFT side
- Gravity side
Kerr Black Holes
• 4d rotating black hole
• Extremal limit: J = M2
• GRS 1915+105:
J » :99M 2
McClintock et al. 2006
• Bekenstein-Hawking
Area Entropy
S
ex t
=
4
= 2¼J
The Kerr/CFT
Correspondence
• The Kerr/CFT correspondence
Introduction
Near-Extremal Kerr/CFT
Black hole decay
- CFT side
- Gravity side
TH, Guica, Song, Strominger '08
Near the horizon of an extremal Kerr black hole, quantum gravity is dual
to a 2D conformal field theory.
Central charge: c = 12 J
• Derivation: states transform under an asymptotic Virasoro algebra
(left-movers only). Gives no details about CFT.
• Application: compute the extremal entropy by counting CFT
microstates using the Cardy formula
• Applies to astrophysical black holes (and more)
– Holographic duality without:
•
•
•
•
•
AdS
Charge
Extra dimensions
Supersymmetry
String theory
The Plan
Introduction
Near-Extremal Kerr/CFT
Black hole decay
- CFT side
- Gravity side
• Review of Kerr/CFT, and some motivation
• Near-extremal black holes
• Black hole decay
Review of Kerr/CFT
Introduction
Near-Extremal Kerr/CFT
Black hole decay
- CFT side
- Gravity side
• Extreme Kerr has an infinite throat, so we can treat the near horizon
³
´
region as its own spacetime
ds2 = 2J f 1 (µ) ¡ r 2 dt 2 +
dr 2
r2
+ dµ2 + f 2 (µ)(dÁ + r dt) 2
Bardeen, Horowitz ‘99
AdS2
• Isometry group SL (2; R)
R
£ U(1) L
• Boundary cond.  Asymptotic symmetry group: Virasoro with
central charge Virasoro
c = 12J
L
L
st law
• Temperature
from
the
1
TL dS ´ dJ ; S = 2¼J
) TL =
1
2¼
• Bekenstein-Hawking entropy
from the CFT via Cardy formula
2
SC F T =
¼
3
cL TL = 2¼J = Sgr av
Introduction
Near-Extremal Kerr/CFT
Black hole decay
- CFT side
- Gravity side
Generalizations to other
extremal black holes
•
4d Kerr
•
Higher dimensions
Guica, TH, Song, Strominger
Lu, Mei, Pope
– multiple U(1)'s
•
Asymptotic (A)dS
– 2 CFTs
•
Charge
– c = 12 J, or c = 6 Q3
•
String theory and Supergravity
– 6d black string (D1-D5-P)
•
Higher Derivative Corrections
Lu, Mei, Pope; TH, Murata, Nishioka, Strominger;
various others
TH, Murata, Nishioka, Strominger
Azeyanagi, Ogawa, Terashima
Nakayama
Chow, Cvetic, Lu, Pope
Lu, Mei, Pope, Vazquez-Poritz
Chen, Wang
etc.
Krishnan, Kuperstein
Azeyanagi, Compere, Ogawa,,
Tachikawa, Terashima
Introduction
Near-Extremal Kerr/CFT
Black hole decay
- CFT side
- Gravity side
Holography for black
holes in the sky
•
Now back to 4d Kerr black holes. Our goal is to apply holography to these
real-world black holes. (For this to be sensible, Kerr/CFT must be extended
at least to near-extremal black holes.)
•
What can we learn about the CFT from gravity, and vice-versa?
•
We need to fill in the holographic dictionary
4d black hole
2d CFT
•Decay
•???
•Scattering
•???
•Bekenstein-Hawking
Entropy
•CFT Microstate
counting
•Hawking radiation
•???
•etc.
•???
The Plan
Introduction
Near-Extremal Kerr/CFT
Black hole decay
- CFT side
- Gravity side
• Review of Kerr/CFT, and some motivation
• Near-extremal black holes
– right-movers and left-movers
– entropy from counting microstates
• Black hole decay
– superradiant emission – also interesting to astrophysicists
– gravity computation
– CFT interpretation
Near-extremal Kerr
Introduction
Near-Extremal Kerr/CFT
Black hole decay
- CFT side
- Gravity side
• Near horizon symmetries in Extremal Kerr/CFT
Exact:
Asymptotic:
U(1) L £ SL (2; R) R
Virasoro £
???
cL = 12J
TL = 1=2¼
cR = ???
TR = 0
• Left-movers TL , cL account for extremal entropy. What about rightmovers?
• L0R = M2 – J = deviation from extremality
• So right-movers with TR , cR should account for near-extremal
entropy.
Near-extremal entropy
Introduction
Near-Extremal Kerr/CFT
Black hole decay
- CFT side
- Gravity side
• Diffeomorphism anomaly
of the
anomaly
/ boundary
c ¡ c theory
=? 0
L
) cR = cL = 12J
R
(?)
r
• Then the CFT entropy with right-movers
excited is (from the Cardy
c
formula)
S
= 2¼J + 2¼ R E + ¢¢¢
CF T
6
R
ER = M 2 ¡ J
• This exactly matches near-extremal Bekenstein-Hawking entropy
– 4d near-extremal Kerr-Newman-AdS black holes
– 5d near-extremal rotating 3-charge black holes (D1-D5-P)
• Summary: We have only derived left-movers from the asymptotic
symmetries, but expect right-movers account for excitations above
extremality (compare: cL, cR in warped AdS)
Introduction
Near-Extremal Kerr/CFT
Black hole decay
- CFT side
- Gravity side
GRS 1915+ 105
J » 0:99M 2
• Matching the near-extremal entropy is evidence that
Kerr/CFT applies to near-extremal astrophysical black
holes like GRS 1915.
• Now on to black hole decay / superradiance
– Energy extraction by classical superradiance; black hole decay
by quantum superradiance
Introduction
Near-Extremal Kerr/CFT
Black hole decay
- CFT side
- Gravity side
Superradiance
• Classical
Press & Teukolsky 1974
scalar ¯eld
Á = e¡ i ! t + i m Á f (r; µ)
¾absor p < 0
• Classical stimulated emission
 quantum spontaneous emission
Movie/image credits:
NASA website
• Quantum
¡ decay =
1
e¡
( ! ¡ m ) =T H
¡ 1
¾absor p
! = energy of mode
m = angular moment um of mode
= black hole rot at ional velocity
at extremality, computation of decay rate = computation of greybody factor
Introduction
Near-Extremal Kerr/CFT
Black hole decay
- CFT side
- Gravity side
Superradiance
from the CFT perspective
Maldacena & Strominger '98
Z
M »
dx + ei !
x+
h¯naljOjinit iali
X
¡ decay
=
jM j 2
»
¯nal
Z
dx + hO(x + )O(0)i ei ( !
»
moment um-space 2-pt funct ion
¡ m )x+
greybody factor = 2-point function in the dual CFT
Not determined by conformal invariance!
Near horizon
gravity
³ perspective
ds2 = 2J f 1 (µ) ¡ r 2 dt 2 +
dr 2
r2
Introduction
Near-Extremal Kerr/CFT
Black hole decay
- CFT side
- Gravity side
´
+ dµ2 + f 2 (µ)(dÁ + r dt) 2
AdS2
• Dimensionally reduce to map superradiance on Kerr to
Schwinger pair production in an electric field on AdS2
• The pair production threshold is
±2
• In 4d language,
´
charge2
±2
=
¡
2m 2
mass2
¡ K `m
1Pioline & Troost; Kim & Page
¡
> 0
4
1
¡
4
m = angular momentum
K = spheroidal harmonic eigenvalue (in 4d, numerical only)
Schwinger Pair
Production on AdS2
R (reflected)
Introduction
Near-Extremal Kerr/CFT
Black hole decay
- CFT side
- Gravity side
Pair product ion
rat e
T (transmitted)
¡ = jTj 2
1 (ingoing)
r = 0 (horizon)
r = 1 (boundary)
Introduction
Near-Extremal Kerr/CFT
Black hole decay
- CFT side
- Gravity side
Schwinger Pair
Production on AdS2
• Scalar wave equation
@r (r 2 @r Á) + [(q + ! =r ) 2 ¡ ¹ 2 ]Á = 0
• AsymptoticÁbehavior
= (r ¡
!
h§
=
h+
+ R! r ¡
h¡
)ei !
t
1
§ i±
2
• Aside: L0R = h for highest weight states (complex
conformal dimension?)
Black hole decay rate
Introduction
Near-Extremal Kerr/CFT
Black hole decay
- CFT side
- Gravity side
• Final result
¡ decay
=
jT! j 2
=
jÁ! (1 )j 2
¡ 1 + e4¼±
1 + e2¼( ±+ m )
=
• This is an extremal limit of the classic formula of Press
and Teukolsky
¡ =
si nh 2 2¼±
cosh 2 ¼( m ¡ ±) + cosh 2 ¼( m + ±) + 2 cos 2¼¾cosh ¼( m + ±) cosh ¼( m ¡ ±)
Relation to CFT
Introduction
Near-Extremal Kerr/CFT
Black hole decay
- CFT side
- Gravity side
• We found the Schwinger
¡ = jÁproduction
(1 )j 2 rate
!
• But this is by definition the
X boundary 2-point function
G( 2) (x) =
Á¤ Á! ei ! x
!
!
• So black hole decay rate is manifestly a CFT 2-point
function, which we just computed. This 2-point function is
not determined by conformal invariance, but is a probe of
the CFT state
• Fourier transform to CFT position space appears
hopeless – δ is a function of the momentum that is only
known numerically
Conclusion
Introduction
Near-Extremal Kerr/CFT
Black hole decay
- CFT side
- Gravity side
• Some questions
– What does the 2-point function tell us about the state of the
CFT?
– Can learn more from the 6d black string? CFT dual is known
from string theory! (work in progress)
• Summary:
– Gravity on extreme Kerr is a CFT
– Applies to various extreme black holes
– With some extra assumptions, extends to near-extremal black
holes
– Started filling in the holographic dictionary, connecting black hole
superradiance to boundary two-point functions