Abstract
We investigate linear perturbations of spin-s fields in the Kerr-AdS black hole and in its near-horizon geometry (NHEK-AdS), using the Teukolsky master equation and the Hertz potential. In the NHEK-AdS geometry we solve the associated angular equation numerically and the radial equation exactly. Having these explicit solutions at hand, we search for linear mode instabilities. We do not find any (non-)axisymmetric instabilities with outgoing boundary conditions. This is in agreement with a recent conjecture relating the linearized stability properties of the full geometry with those of its near-horizon geometry. Moreover, we find that the asymptotic behaviour of the metric perturbations in NHEK-AdS violates the fall-off conditions imposed in the formulation of the Kerr/CFT correspondence (the only exception being the axisymmetric sector of perturbations).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
B.F. Whiting, Mode stability of the Kerr black hole, J. Math. Phys. 30 (1989) 1301 [INSPIRE].
W.H. Press and S.A. Teukolsky, Perturbations of a Rotating Black Hole. II. Dynamical Stability of the Kerr Metric, Astrophys. J. 185 (1973) 649 [INSPIRE].
S.A. Teukolsky, Perturbations of a rotating black hole. 1. Fundamental equations for gravitational electromagnetic and neutrino field perturbations, Astrophys. J. 185 (1973) 635 [INSPIRE].
J.M. Bardeen and G.T. Horowitz, The Extreme Kerr throat geometry: A Vacuum analog of AdS 2 × S 2, Phys. Rev. D 60 (1999) 104030 [hep-th/9905099] [INSPIRE].
A.J. Amsel, G.T. Horowitz, D. Marolf and M.M. Roberts, No Dynamics in the Extremal Kerr Throat, JHEP 09 (2009) 044 [arXiv:0906.2376] [INSPIRE].
O.J. Dias, H.S. Reall and J.E. Santos, Kerr-CFT and gravitational perturbations, JHEP 08 (2009) 101 [arXiv:0906.2380] [INSPIRE].
D. Marolf, The dangers of extremes, Gen. Rel. Grav. 42 (2010) 2337 [arXiv:1005.2999] [INSPIRE].
S. Aretakis, Horizon Instability of Extremal Black Holes, arXiv:1206.6598 [INSPIRE].
J. Lucietti and H.S. Reall, Gravitational instability of an extreme Kerr black hole, arXiv:1208.1437 [INSPIRE].
B. Carter, Hamilton-Jacobi and Schrödinger separable solutions of Einstein’s equations, Commun. Math. Phys. 10 (1968) 280 [INSPIRE].
O.J. Dias, G.T. Horowitz and J.E. Santos, Gravitational Turbulent Instability of Anti-de Sitter Space, Class. Quant. Grav. 29 (2012) 194002 [arXiv:1109.1825] [INSPIRE].
S. Hawking and H. Reall, Charged and rotating AdS black holes and their CFT duals, Phys. Rev. D 61 (2000) 024014 [hep-th/9908109] [INSPIRE].
V. Cardoso, O.J. Dias and S. Yoshida, Classical instability of Kerr-AdS black holes and the issue of final state, Phys. Rev. D 74 (2006) 044008 [hep-th/0607162] [INSPIRE].
H. Lü, J. Mei and C. Pope, Kerr/CFT Correspondence in Diverse Dimensions, JHEP 04 (2009) 054 [arXiv:0811.2225] [INSPIRE].
M. Durkee and H.S. Reall, Perturbations of near-horizon geometries and instabilities of Myers-Perry black holes, Phys. Rev. D 83 (2011) 104044 [arXiv:1012.4805] [INSPIRE].
O.J. Dias, R. Monteiro, H.S. Reall and J.E. Santos, A Scalar field condensation instability of rotating anti-de Sitter black holes, JHEP 11 (2010) 036 [arXiv:1007.3745] [INSPIRE].
N. Tanahashi and K. Murata, Instability in near-horizon geometries of even-dimensional Myers-Perry black holes, arXiv:1208.0981 [INSPIRE].
B. Chen and J. Long, On Holographic description of the Kerr-Newman-AdS-dS black holes, JHEP 08 (2010) 065 [arXiv:1006.0157] [INSPIRE].
M. Guica, T. Hartman, W. Song and A. Strominger, The Kerr/CFT Correspondence, Phys. Rev. D 80 (2009) 124008 [arXiv:0809.4266] [INSPIRE].
G. Compère, The Kerr/CFT correspondence and its extensions: a comprehensive review, arXiv:1203.3561 [INSPIRE].
M. Godazgar, The perturbation theory of higher dimensional spacetimes a la Teukolsky, Class. Quant. Grav. 29 (2012) 055008 [arXiv:1110.5779] [INSPIRE].
M. Guica and A. Strominger, Microscopic Realization of the Kerr/CFT Correspondence, JHEP 02 (2011) 010 [arXiv:1009.5039] [INSPIRE].
I. Bena, M. Guica and W. Song, Un-twisting the NHEK with spectral flows, arXiv:1203.4227 [INSPIRE].
G. Gibbons, M. Perry and C. Pope, The First law of thermodynamics for Kerr-anti-de Sitter black holes, Class. Quant. Grav. 22 (2005) 1503 [hep-th/0408217] [INSPIRE].
M.M. Caldarelli, G. Cognola and D. Klemm, Thermodynamics of Kerr-Newman-AdS black holes and conformal field theories, Class. Quant. Grav. 17 (2000) 399 [hep-th/9908022] [INSPIRE].
M.M. Caldarelli, O.J. Dias and D. Klemm, Dyonic AdS black holes from magnetohydrodynamics, JHEP 03 (2009) 025 [arXiv:0812.0801] [INSPIRE].
S. Teukolsky and W. Press, Perturbations of a rotating black hole. III - Interaction of the hole with gravitational and electromagnet ic radiation, Astrophys. J. 193 (1974) 443 [INSPIRE].
C.M. Chambers and I.G. Moss, Stability of the Cauchy horizon in Kerr-de Sitter space-times, Class. Quant. Grav. 11 (1994) 1035 [gr-qc/9404015] [INSPIRE].
M. Giammatteo and I.G. Moss, Gravitational quasinormal modes for Kerr anti-de Sitter black holes, Class. Quant. Grav. 22 (2005) 1803 [gr-qc/0502046] [INSPIRE].
R.A. Breuer, M.P. Ryan and S. Waller, Some Properties of Spin-Weighted Spheroidal Harmonics, Proc. R. Soc. Lond. A 22 (1977) 71.
E. Berti, V. Cardoso and M. Casals, Eigenvalues and eigenfunctions of spin-weighted spheroidal harmonics in four and higher dimensions, Phys. Rev. D 73 (2006) 024013 [Erratum ibid. D 73 (2006) 109902] [gr-qc/0511111] [INSPIRE].
H. Suzuki, E. Takasugi and H. Umetsu, Analytic solutions of Teukolsky equation in Kerr-de Sitter and Kerr-Newman-de Sitter geometries, Prog. Theor. Phys. 102 (1999) 253 [gr-qc/9905040] [INSPIRE].
A. Strominger, AdS 2 quantum gravity and string theory, JHEP 01 (1999) 007 [hep-th/9809027] [INSPIRE].
M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical tables, Applied Mathematics Series, U.S. Govt. Print. Off. (1964).
J. Friedman, Ergosphere instability, Commun. Math. Phys. 63 (1978) 243.
J.M. Cohen and L.S. Kegeles, Space-time Perturbations, Phys. Lett. A 54 (1975) 5.
L. Kegeles and J. Cohen, Constructive procedure for perturbations of space-times, Phys. Rev. D 19 (1979) 1641 [INSPIRE].
P. Chrzanowski, Vector Potential and Metric Perturbations of a Rotating Black Hole, Phys. Rev. D 11 (1975) 2042 [INSPIRE].
J.M. Stewart, Hertz-Bromowich-Debye-Whittaker-Penrose Potentials in General Relativity, Proc. Roy. Soc. Lond. A 367 (1979) 527.
R.M. Wald, Construction of Solutions of Gravitational, Electromagnetic, Or Other Perturbation Equations from Solutions of Decoupled Equations, Phys. Rev. Lett. 41 (1978) 203 [INSPIRE].
L.R. Price, K. Shankar and B.F. Whiting, On the existence of radiation gauges in Petrov type-II spacetimes, Class. Quant. Grav. 24 (2007) 2367 [gr-qc/0611070] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1208.3322
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Dias, Ó.J.C., Santos, J.E. & Stein, M. Kerr-AdS and its near-horizon geometry: perturbations and the Kerr/CFT correspondence. J. High Energ. Phys. 2012, 182 (2012). https://doi.org/10.1007/JHEP10(2012)182
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2012)182