black hole decay in the kerr/cft correspondence
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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. 2. J. M. 9. 9. A. ». r. e. a. S. J. 2. :. ¼. =. - PowerPoint PPT PresentationTRANSCRIPT
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