black hole decay in the kerr/cft correspondence 0809.4266 th, guica, song, and strominger and work...

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


• 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


The Plan

• Review of Kerr/CFT, and some motivation

• Near-extremal black holes

• Black hole decay


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 = ¼2

3cL TL = 2¼J = Sgrav

TL dS ´ dJ ; S = 2¼J ) TL = 12¼


• 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


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 CFT

4d black hole

•Decay

•Scattering

•Bekenstein-Hawking Entropy

•Hawking radiation

•etc.

•???

•???

•CFT Microstate counting

•???

•???


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


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


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 +2¼

rcR6ER +¢¢¢

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


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


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


r = 1 (boundary)r = 0 (horizon)

Schwinger PairProduction on AdS2

1 (ingoing)

T (transmitted)

R (reflected)

Pair productionrate

¡ = jTj2


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±


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¡ ±)


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


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

black hole decay in the kerr/cft correspondence 0809.4266 th, guica, song, and strominger and work...

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