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Superradiance and Stability of Higher-Dimensional Black Holes

Superradiance and Stability of Higher-Dimensional Black Holes. Hideo Kodama Cosmophysics Group IPNS, KEK. Present Status of the BH Stability Issue. Four-Dimensional Black Holes Stable Schwarzschild black hole [Vishveshwara 1970; Price 1972; Wald 1979,1980]

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Superradiance and Stability of Higher-Dimensional Black Holes

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  1. Superradiance and Stability of Higher-Dimensional Black Holes Hideo Kodama Cosmophysics Group IPNS, KEK

  2. Present Status of the BH Stability Issue Four-Dimensional Black Holes • Stable • Schwarzschild black hole [Vishveshwara 1970; Price 1972; Wald 1979,1980] • Reissner-Nordstrom black hole [Chandrasekhar 1983] • AdS/dS (charged) black holes [Ishibashi, Kodama 2003, 2004] • Kerr black hole [Whiting 1989] • Skyrme black hole (non-unique system) [Heusler, Droz, Straumann 1991,1992; Heusler, Straumann, Zhou 1993] • Unstable • YM black hole (non-unique system) [Straumann, Zhou 1990; Bizon 1991; Zhou, Straumann 1991] • Unknown • Kerr-Newman black hole • Conjecture: large AdS-Kerr black holes are stable, but small ones are SR unstable[Hawking, Reall 1999; Cardoso, Dias 2004]

  3. Higher-Dimensional Black Objects • Stable • AF vacuum static (Schwarzschild-Tangherlini) [IK 2003] • AF charged static (D=5,6-11) [IK 2004; Konoplya, Zhidenko 2007] • dS vacuum static (D=5,6,7-11), dS charged static (D=5,6-11) [IK 2003, 2004;Konoplya, Zhidenko 2007] • BPS charged black branes (in type II SUGRA) [Gregory, Laflamme 1994:Hirayama, Kang, Lee 2003] • Unstable • Static black string (in AdS bulk), black branes (non-BPS) [Gregory, Laflamme 1993, 1995; Gregory 2000; Hirayama, Kang 2001: Hirayama, Kang, Lee 2003; Kang; Seahra, Clarkson, Maartens 2005; Kudoh 2006] • Rapidly rotating special GLPP (AdS-Kerr) bh[Kunhuri, Lucietti, Reall 2006] • Unknown • AF charged static (D>11) • AdS (charged) static (D>4) • dS (charged) static (D>11) • Conjecture: black rings are GL unstable • Conjecture: rapidly rotating MP black holes are GL unstable [Emparan, Myers 2003] • Conjecture: doubly spinning black rings are SR unstable[Dias 2006] • Conjecture:Kerr black brane Kerr4£ Rp is SR unstable[Cardoso, Yoshida 2005]

  4. Superradiance

  5. DOC AF Stationary Rotating Black Hole • Global Structure • Symmetries • Rotation

  6. Asymptotic Structure • At infinity where f(r)=1-2M/rn-1and n=D-2. • Near horizon This metric can be written in a regular form in terms of the advanced/retarded time coordinate

  7. Massless Scalar Field • Klein-Gordon product From the field equation the KG product defined by is independent of the choice of the Cauchy surface  in DOC. • Scattering problem No incoming wave from the black hole

  8. Asymptotic behaviour • At infinity • At horizon where *= – mi ih.

  9. Superradiance • Flux conservation • Superradiance condition This condition is equivalent to Cf. Penrose process in the ergo region [Penrose 1969]

  10. Black Hole Bombs • Black hole in a mirror box [Zel’dovich 1971; Press, Teukolsky 1972; Cardoso, Dias, Lemos, Yoshida 2004] • Massive bosonic fields around a black hole [Damour, Deruelle, Ruffini 1976;] For astrophysical black holes, the growth rate of the instability is negligibly small [Zouros, Eardley 1979], but for small primordial black holes, it may become comparable to the dynamical times scale [Detweiler 1980; Furuhashi, Nambu 2004; Dolan 2007] • Spinning black strings[Marolf, Palmer 2004; Cardoso, Lemos 2005; Cardoso, Yoshida 2005] ; Doubly spinning black rings[Dias 2006] Kerr4£ Rp⇒ KK modes provide effective masses in 4D. • AdS-Kerr black holes[Hawking, Reall 1999; Cardoso, Dias 2004; Cardoso, Dias, Yoshida 2006] Cf. Magnetic Penrose process and relativistic cosmic jets in GRB [van Putten 2000; Aguirre 2000; Nagataki, Takahashi, Mizuta, Tachiwaki 2007]

  11. Instability of AdS-Kerr Black Holes

  12. DOC AdS-Kerr Black Holes • Global Structure The infinity is time-like and DOC is not globally hyperbolic. • Symmetries The time-translation Killing vector is not unique. • Rotation The structure near the horizon is the case as in the AF case. But the angular velocity h of black hole depends on the choice of .

  13. Superradiance in AdS-Kerr BH? • Flux Conservation • Asymptotic behavior of a massless scalar field at infinity: for r»1, Therefore, Ref: Hawking SW, Real HS: PRD61, 024014 (1999)

  14. Cautions • Mass of a scalar field in AdS spacetime The free field equation can be written at r»1 (x»1) as in terms of the variables Hence, taking account of the redshift factor for the frequency , the Breitenloehner-Freedman bound 2=-(n+1)2/4 corresponds to the massless field in the AF case. In other words, for 2=0, the scalar field is effectively massive.

  15. Slowly Rotating AdS-Kerr • Everywhere Time-like Killing Vector For slowly rotating black hole, there exists a Killing vector that is everywhere timelike in DOC: for example, when ai2 l2< rh4 (i=1,2) for D=5, or when a12l2< rh4, a2=…=aN=0. • Energy Conservation Law In this case, no instability occurs for a matter field satisfying the dominant energy condition [Hawking, Reall 1999] where n T k is non-negative everywhere on . • Stability Conjecture On the basis of this observation, Hawking and Reall conjectured that AdS-Kerr black holes with slow rotations such that ai2 l2< rh4 will be stable against gravitational perturbations as well. At the same time, they also conjecture that rapidly rotating AdS-Kerr black holes will be unstable. This conjecture was proved for D=4 and rh¿l[Cardoso, Dias, Yoshida 2006]

  16. Tensor Modes in Special AdS-Kerr • Simple AdS-Kerr: a1=a, a2=…=aN=0 • In this case, the metric is U(1)£ SO(n+1) symmetric with n=D-4. • For D¸ 7, the harmonic amplitude HT for tensor-type metric perturbations obeys the equation This equation is exactly identical to the equation for the harmonic amplitude for a minimally-coupled massless scalar field in the same background! Therefore, we can apply the results on stability/instability of a massless scalar field to the tensor modes. • In particular, we can conclude that tensor perturbations are stable for a2l2 < rh4 on the basis of the argument by Hawking, Reall 1999.

  17. Energy Integral for Tensor Perturbations of Simple AdS-Kerr: In the coordinates in which the metric is written for (t,r,x) defined by the following energy integral is conserved: where , F and U0 are always positive outside horizon, while U1 is positive definite only for a2l 2 < rh4.

  18. Effective Potential • In the effective potential both U0 and U1 are positive for a2l 2 < rh4. • For a2l 2 > rh4 , however, U1 becomes negative in some range of r at x=-1, and the negative dip of the potential becomes arbitrarily deep as m increases. • Hence, it is highly probable that simple AdS-Kerr black holes in dimensions higher than 6 are unstable for tensor perturbations. • If we take ! 0 ( l !1) limit with fixed a and rh, the above stability condition is violated. This may suggest the instability of MP black holes unless the growth rate of instability vanishes at this limit.

  19. Equally Rotating AdS-Kerr: a1==aN=a with D=2N+1. • In this case, the angular part of the metric has the structure of a twisted S1 bundle over CPN-1. • For a special class of tensor perturbations, the metric perturbation equation can be reduced to a Schrodinger-type ODE that has the same structure as that for a massless free scalar field. • It is claimed on the basis of analysis utilising the WKB approximation that such tensor perturbations satisfying the “superradiant condition” =m h are unstable if hl > 1, i.e., if there does not exist a global timelike Killing vector. [Kunhuri, Lucietti, Reall 2006]

  20. Summary • Superradiance • Superradiance is a quite universal phenomenon independent of spacetime dimensions for an asymptotically flat black hole with charge or rotation. • It occurs when there is a particle state at infinity that becomes unphysical at horizon. • Superradiance may provoke various instabilities in systems containing black holes in both four and higher dimensions.

  21. AdS-Kerr black holes • It is now quite likely that adS-Kerr black holes are unstable in four and higher dimensions when the rotation velocity at infinity lh exceed the light velocity while they are stable in the opposite situation that includes the case rh> l . • Many authors argue thatthis instability is caused by superradiance, but it is not so clear. • In the simply rotating case, the equation for tensor-type perturbations is identical to that for a free massless scalar field and separable to decoupled ODEs. For this case, we will be able to draw a clear conclusion on the stability issue by numerical calculations including the AF case. • In this relation, it will be also interesting to study the stability problem of a free scalar field around a higher-dimensional black hole. • The final state of the instability is a very important problem to be studied next.

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