Black Hole Decay in the Kerr/CFT Correspondence

1. 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. 2 J M 9 9 A » r e a S J 2 : ¼ = = t 4 e x Kerr Black Holes • 4d rotating black hole • Extremal limit: J = M2 • GRS 1915+105: • Bekenstein-Hawking Entropy McClintock et al. 2006

3. The Kerr/CFT Correspondence • The Kerr/CFT correspondence 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 TH, Guica, Song, Strominger '08

4. The Plan • Review of Kerr/CFT, and some motivation • Near-extremal black holes • Black hole decay

5. d A S 2 ³ ´ 2 d 2 2 2 2 2 ( ) ( ) ( ) d f µ d r d µ f µ d Á d 1 J 2 t t ¡ + + + + d d T S J S J T 2 s r r = ( ) ( ) 1 2 ´ S V L R J U i ¼ 2 1 2 1 2 = ) = 2 £ L L c r a s o r o r = ; 2 L L R L ¼ S T J S 2 ¼ ; c ¼ = = = C F T L L Bardeen, Horowitz ‘99 g r a v 3 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

6. Generalizations to otherextremal black holes • 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 Guica, TH, Song, Strominger Lu, Mei, Pope 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

7. 2d CFT 4d black hole • Decay • Scattering • Bekenstein-Hawking Entropy • Hawking radiation • etc. • ??? • ??? • CFT Microstate counting • ??? • ??? 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

8. 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

9. ? ? ? J 1 2 c c Exact: = = L R ( ) ( ) U S L R 1 2 = £ T T 1 0 2 ¼ L R = = L R ; ? ? ? V i £ r a s o r o Asymptotic: 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.

10. ? l 0 ¡ a n o m a y / c c = L R r c R ( ) ? J 1 2 S J E c c 2 2 ) = = + + R L ¢ ¢ ¢ ¼ ¼ = C F T R 6 2 E M J ¡ = R 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)

11. G R S 1 9 1 5 1 0 5 + 2 J M 0 9 9 » : • 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

12. 1 l ¯ l d s c a a r e ¡ ¾ = d b a s o r p ( ) = e c a y ­ T ¡ ¡ Á i i 1 ¡ + t ! m ¡ ( ) H Á f µ ! m e e r = ; 0 f d < ¾ b ! e n e r g y o m o e = a s o r p l f d t m a n g u a r m o m e n u m o m o e = b l k h l l l ­ i i t t t a c o e r o a o n a v e o c y = Superradiance • Classical • Classical stimulated emission  quantum spontaneous emission • Quantum at extremality, computation of decay rate = computation of greybody factor Press & Teukolsky 1974 Movie/image credits: NASA website

13. Z + i + h j j i ! x d ¯ l l M O i i i t x e n a n a » X 2 j j ¡ M = d e c a y ¯ l n a Z + ( ) ­ i + + ¡ h ( ) ( ) i ! m x d O O 0 x x e » f i 2 t t t m o m e n u m - s p a c e - p u n c o n » Superradiance from the CFT perspective Maldacena & Strominger '98 greybody factor = 2-point function in the dual CFT Not determined by conformal invariance!

14. d A S 1 1 2 2 2 2 2 2 ³ ´ ± ± h K 2 0 ¡ ¡ ¡ ¡ 2 > ´ c a r g e m m a s s d = ` 2 2 2 2 2 ( ) ( ) ( ) d f µ d r d µ f µ d Á d J m 2 t t ¡ + + + + 4 4 s r r = 1 2 2 r Pioline & Troost; Kim & Page 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)

15. ( ( ) ) h b d i 0 r r 1 o r o u z n o n a r y = = d P i i t a r p r o u c o n t r a e 2 j j ¡ T = Schwinger PairProduction on AdS2 R (reflected) T (transmitted) 1 (ingoing)

16. h h i ¡ ¡ t ( ) ! Á R + ¡ + r r e = 2 2 2 ! ! ( ) [ ( = ) ] @ @ Á Á 0 + + ¡ r q ! r ¹ = r r 1 h ± i § = § 2 Schwinger PairProduction on AdS2 • Scalar wave equation • Asymptotic behavior • Aside: L0R = h for highest weight states (complex conformal dimension?)

17. 2 2 j j h ± ¡ T i 2 s n ¼ ¡ = d = ! e c a y 2 2 ( ) ( ) ( ) ( ) h ± h ± h ± h ± 2 2 ¡ + + + + ¡ c o s ¼ m c o s ¼ m c o s ¼ ¾ c o s ¼ m c o s ¼ m 2 j ( ) j Á 1 = ! ± 4 ¼ 1 ¡ + e = ( ) ± 2 + 1 ¼ m + e Black hole decay rate • Final result • This is an extremal limit of the classic formula of Press and Teukolsky

18. X ( ) i 2 ( ) ¤ ! x Á Á G x e = ! 2 ! j ( ) j Á ¡ 1 = ! ! 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

19. 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