1. Black Hole Decay in the Kerr/CFT Correspondence 0809.4266 TH, Guica, Song, and Strominger and work in progress with Song and Strominger ESI Workshop on Gravity in Three Dimensions April 2009 Tom Hartman Harvard University

2. Introduction Near-Extremal Kerr/CFT Black hole decay - CFT side - Gravity side Kerr Black Holes 4d rotating black hole Extremal limit: J = M 2 GRS 1915+105: Bekenstein-Hawking Entropy S ex t = A rea 4 = 2 ¼ J McClintock et al. 2006 J » : 99 M 2

3. Introduction Near-Extremal Kerr/CFT Black hole decay - CFT side - Gravity side 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 The Kerr/CFT Correspondence TH, Guica, Song, Strominger '08

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

5. Introduction Near-Extremal Kerr/CFT Black 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 1 st law Bekenstein-Hawking entropy from the CFT via Cardy formula SL ( 2 ; R ) R £ U ( 1 ) L V i rasoro L c L = 12 J Bardeen, Horowitz ‘99 A d S 2 d s 2 = 2 J f 1 ( µ ) ³ ¡ r 2 d t 2 + d r 2 r 2 + dµ 2 + f 2 ( µ )( dÁ + r d t ) 2 ´ S CFT = ¼ 2 3 c L T L = 2 ¼ J = S grav T L d S ´ d J ; S = 2 ¼ J ) T L = 1 2 ¼

6. Introduction Near-Extremal Kerr/CFT Black 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 Q 3 String theory and Supergravity –6d black string (D1-D5-P) Higher Derivative Corrections Generalizations to other extremal black holes Guica, TH, Song, Strominger Lu, Mei, Pope TH, Murata, Nishioka, Strominger Azeyanagi, Ogawa, Terashima Nakayama Chow, Cvetic, Lu, Pope Lu, Mei, Pope, Vazquez-Poritz Chen, Wang etc. Lu, Mei, Pope; TH, Murata, Nishioka, Strominger; various others Krishnan, Kuperstein Azeyanagi, Compere, Ogawa,, Tachikawa, Terashima

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

8. Introduction Near-Extremal Kerr/CFT Black 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

9. Introduction Near-Extremal Kerr/CFT Black hole decay - CFT side - Gravity side Near-extremal Kerr Near horizon symmetries in Extremal Kerr/CFT Left-movers T L, c L account for extremal entropy. What about right- movers? L 0 R = M 2 – J = deviation from extremality So right-movers with T R, c R should account for near-extremal entropy. Exact: Asymptotic: U ( 1 ) L £ SL ( 2 ; R ) R V i rasoro £ ??? c L = 12 J T L = 1 = 2 ¼ c R = ??? T R = 0

10. Introduction Near-Extremal Kerr/CFT Black 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: c L, c R in warped AdS) S CFT = 2 ¼ J + 2 ¼ r c R 6 E R + ¢¢¢ E R = M 2 ¡ J anoma l y/c L ¡ c R ? = 0 ) c R = c L = 12 J ( ? )

11. Introduction Near-Extremal Kerr/CFT Black hole decay - CFT side - Gravity side GRS 1915 + 105 J » 0 : 99 M 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

12. Introduction Near-Extremal Kerr/CFT Black hole decay - CFT side - Gravity side Superradiance Movie/image credits: NASA website Classical Classical stimulated emission  quantum spontaneous emission Quantum at extremality, computation of decay rate = computation of greybody factor sca l ar ¯ e ld Á = e ¡ i ! t + i m Á f ( r ; µ ) ¾ a b sorp < 0 ¡ d ecay = 1 e ¡ ( ! ¡ m ­ )= T H ¡ 1 ¾ a b sorp ! = energyo f mo d e m = angu l armomen t umo f mo d e ­ = bl ac kh o l ero t a t i ona l ve l oc i t y Press & Teukolsky 1974

13. Introduction Near-Extremal Kerr/CFT Black hole decay - CFT side - Gravity side Superradiance from the CFT perspective M » Z d x + e i !x + h ¯ na l j O j i n i t i a l i ¡ d ecay = X ¯ na l j M j 2 » Z d x + h O ( x + ) O ( 0 )i e i ( ! ¡ m ­ ) x + » momen t um-space 2 -p t f unc t i on greybody factor = 2-point function in the dual CFT Not determined by conformal invariance! Maldacena & Strominger '98

14. Introduction Near-Extremal Kerr/CFT Black 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 AdS 2 The pair production threshold is In 4d language, m = angular momentum K = spheroidal harmonic eigenvalue (in 4d, numerical only) A d S 2 d s 2 = 2 J f 1 ( µ ) ³ ¡ r 2 d t 2 + d r 2 r 2 + dµ 2 + f 2 ( µ )( dÁ + r d t ) 2 ´ Pioline & Troost; Kim & Page ± 2 ´ c h arge 2 ¡ mass 2 ¡ 1 4 > 0 ± 2 = 2 m 2 ¡ K ` m ¡ 1 4

15. Introduction Near-Extremal Kerr/CFT Black hole decay - CFT side - Gravity side r = 1 ( b oun d ary ) r = 0 ( h or i zon ) Schwinger Pair Production on AdS 2 1 (ingoing) T (transmitted) R (reflected) P a i rpro d uc t i on ra t e ¡ = j T j 2

16. Introduction Near-Extremal Kerr/CFT Black hole decay - CFT side - Gravity side Schwinger Pair Production on AdS 2 Scalar wave equation Asymptotic behavior Aside: L 0 R = h for highest weight states (complex conformal dimension?) @ r ( r 2 @ r Á ) + [( q + ! = r ) 2 ¡ ¹ 2 ] Á = 0 Á ! = ( r ¡ h + + R ! r ¡ h ¡ ) e i ! t h § = 1 2 § i ±

17. Introduction Near-Extremal Kerr/CFT Black 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 ¡ d ecay = j T ! j 2 = j Á ! ( 1 )j 2 = ¡ 1 + e 4 ¼ ± 1 + e 2 ¼ ( ± + m ) ¡ = s i n h 2 2 ¼ ± cos h 2 ¼ ( m ¡ ± ) + cos h 2 ¼ ( m + ± ) + 2 cos 2 ¼¾cos h ¼ ( m + ± ) cos h ¼ ( m ¡ ± )

18. Introduction Near-Extremal Kerr/CFT Black 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 ¡ = j Á ! ( 1 )j 2 G ( 2 ) ( x ) = X ! Á ¤ ! Á ! e i !x

19. Introduction Near-Extremal Kerr/CFT Black 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