Black Hole Decay in the Kerr/CFT Correspondence
0809.4266 TH, Guica, Song, and Stromingerand work in progress with Song and Strominger
ESI Workshop on Gravity in Three DimensionsApril 2009
Tom HartmanHarvard University
IntroductionNear-Extremal Kerr/CFTBlack hole decay - CFT side - Gravity side
Kerr Black Holes• 4d rotating black hole
• Extremal limit: J = M2
• GRS 1915+105:
• Bekenstein-Hawking Entropy
Sext = Area4 =2¼J
McClintock et al. 2006
J » :99M 2
IntroductionNear-Extremal Kerr/CFTBlack hole decay - CFT side - Gravity side
• The Kerr/CFT correspondenceNear 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 Kerr/CFT Correspondence
TH, Guica, Song, Strominger '08
IntroductionNear-Extremal Kerr/CFTBlack hole decay - CFT side - Gravity side
The Plan• Review of Kerr/CFT, and some motivation
• Near-extremal black holes
• Black hole decay
IntroductionNear-Extremal Kerr/CFTBlack hole decay - CFT side - Gravity side
Review of Kerr/CFT• Extreme Kerr has an infinite throat, so we can treat the near horizon
region as its own spacetime
• Isometry group
• Boundary cond. Asymptotic symmetry group: Virasoro with central charge
• Temperature from the 1st law
• Bekenstein-Hawking entropy from the CFT via Cardy formula
SL(2;R)R £ U(1)L
VirasoroL cL = 12J
Bardeen, Horowitz ‘99AdS2
ds2 = 2J f 1(µ)³¡ r2dt2+ dr 2
r 2 +dµ2+f 2(µ)(dÁ+rdt)2´
SC F T = ¼23 cL TL = 2¼J = Sgrav
TL dS ´ dJ ; S = 2¼J ) TL = 12¼
IntroductionNear-Extremal Kerr/CFTBlack hole decay - CFT side - Gravity side
• 4d Kerr
• Higher dimensions– 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
Generalizations to otherextremal black holes
Guica, TH, Song, Strominger
Lu, Mei, Pope
TH, Murata, Nishioka, Strominger
Azeyanagi, Ogawa, TerashimaNakayamaChow, Cvetic, Lu, PopeLu, Mei, Pope, Vazquez-PoritzChen, Wangetc.
Lu, Mei, Pope; TH, Murata, Nishioka, Strominger; various others
Krishnan, KupersteinAzeyanagi, Compere, Ogawa,, Tachikawa, Terashima
IntroductionNear-Extremal Kerr/CFTBlack 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 dictionary2d CFT4d black hole
•Decay
•Scattering
•Bekenstein-Hawking Entropy
•Hawking radiation
•etc.
•???
•???
•CFT Microstate counting
•???
•???
IntroductionNear-Extremal Kerr/CFTBlack hole decay - CFT side - Gravity side
The Plan• 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
IntroductionNear-Extremal Kerr/CFTBlack hole decay - CFT side - Gravity side
Near-extremal Kerr• Near horizon symmetries in Extremal Kerr/CFT
• Left-movers TL , cL account for extremal entropy. What about right-movers?
• L0R = M2 – J = deviation from extremality
• So right-movers with TR , cR should account for near-extremal entropy.
Exact:
Asymptotic:
U(1)L £ SL(2;R)R
Virasoro £ ???cL =12JTL =1=2¼
cR =???TR =0
IntroductionNear-Extremal Kerr/CFTBlack hole decay - CFT side - Gravity side
Near-extremal entropy• Diffeomorphism anomaly of the boundary theory
• Then the CFT entropy with right-movers excited is (from the Cardy formula)
• 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)
SC F T = 2¼J +2¼r cR
6 ER +¢¢¢
ER =M 2 ¡ J
anomaly / cL ¡ cR ?= 0) cR = cL = 12J (?)
IntroductionNear-Extremal Kerr/CFTBlack hole decay - CFT side - Gravity sideGRS 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
IntroductionNear-Extremal Kerr/CFTBlack hole decay - CFT side - Gravity sideSuperradiance
Movie/image credits:
NASA website
• Classical
• Classical stimulated emission quantum spontaneous emission
• Quantum
at extremality, computation of decay rate = computation of greybody factor
scalar ¯eldÁ= e¡ i ! t+imÁf (r;µ)¾absorp <0
¡ decay =1
e¡ (! ¡ m )=TH ¡ 1¾absorp! = energy of modem= angular momentumof mode = black hole rotational velocity
Press & Teukolsky 1974
IntroductionNear-Extremal Kerr/CFTBlack hole decay - CFT side - Gravity side
Superradiance from the CFT perspective
M »Zdx+ei ! x+ h̄ naljOjinitiali
¡ decay =X
¯naljM j2
»Zdx+hO(x+)O(0)iei (! ¡ m )x+
» momentum-space2-pt function
greybody factor = 2-point function in the dual CFT
Not determined by conformal invariance!
Maldacena & Strominger '98
IntroductionNear-Extremal Kerr/CFTBlack hole decay - CFT side - Gravity side
Near horizon gravity perspective
• Dimensionally reduce to map superradiance on Kerr to Schwinger pair production in an electric field on AdS2
• The pair production threshold is
• In 4d language,
m = angular momentum K = spheroidal harmonic eigenvalue (in 4d, numerical only)
AdS2
ds2 = 2J f 1(µ)³¡ r2dt2+ dr 2
r 2 +dµ2+f 2(µ)(dÁ+rdt)2´
Pioline & Troost; Kim & Page
±2 ´ charge2 ¡ mass2 ¡ 14 > 0
±2 =2m2 ¡ K `m ¡ 14
IntroductionNear-Extremal Kerr/CFTBlack hole decay - CFT side - Gravity side
r = 1 (boundary)r = 0 (horizon)
Schwinger PairProduction on AdS2
1 (ingoing)
T (transmitted)
R (reflected)Pair productionrate
¡ = jTj2
IntroductionNear-Extremal Kerr/CFTBlack hole decay - CFT side - Gravity side
Schwinger PairProduction on AdS2
• Scalar wave equation
• Asymptotic behavior
• Aside: L0R = h for highest weight states (complex
conformal dimension?)
@r (r2@rÁ) + [(q+! =r)2 ¡ ¹ 2]Á= 0
Á! = (r¡ h+ +R! r¡ h¡ )ei ! t
h§ = 12 § i±
IntroductionNear-Extremal Kerr/CFTBlack hole decay - CFT side - Gravity side
Black hole decay rate
• Final result
• This is an extremal limit of the classic formula of Press and Teukolsky
¡ decay = jT! j2
= jÁ! (1 )j2
= ¡ 1+e4¼±1+e2¼(±+m)
¡ = sinh2 2¼±cosh2 ¼(m¡ ±)+cosh2 ¼(m+±)+2cos2¼¾cosh¼(m+±) cosh¼(m¡ ±)
IntroductionNear-Extremal Kerr/CFTBlack hole decay - CFT side - Gravity side
Relation to CFT• We found the Schwinger production rate
• But this is by definition the boundary 2-point function
• 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
¡ = jÁ! (1 )j2
G(2)(x) =X
!Á¤! Á! ei ! x
IntroductionNear-Extremal Kerr/CFTBlack hole decay - CFT side - Gravity side
Conclusion• 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