Abstract
We show that the emergent near-horizon conformal symmetry of extremal black holes gives rise to universal behavior in perturbing fields, both near and far from the black hole horizon. The scale-invariance of the near-horizon region entails power law time-dependence with three universal features: (1) the decay off the horizon is always precisely twice as fast as the decay on the horizon; (2) the special rates of 1/t off the horizon and \( 1/\sqrt{v} \) on the horizon commonly occur; and (3) sufficiently high-order transverse derivatives grow on the horizon (Aretakis instability). The results are simply understood in terms of near-horizon (AdS2) holography. We first show how the general features follow from symmetry alone and then go on to present the detailed universal behavior of scalar, electromagnetic, and gravitational perturbations of d-dimensional electrovacuum black holes.
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References
J.M. Maldacena, J. Michelson and A. Strominger, Anti-de Sitter fragmentation, JHEP 02 (1999) 011 [hep-th/9812073] [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].
H.K. Kunduri, J. Lucietti and H.S. Reall, Near-horizon symmetries of extremal black holes, Class. Quant. Grav. 24 (2007) 4169 [arXiv:0705.4214] [INSPIRE].
P. Figueras, H.K. Kunduri, J. Lucietti and M. Rangamani, Extremal vacuum black holes in higher dimensions, Phys. Rev. D 78 (2008) 044042 [arXiv:0803.2998] [INSPIRE].
M. Guica, T. Hartman, W. Song and A. Strominger, The Kerr/CFT Correspondence, Phys. Rev. D 80 (2009) 124008 [arXiv:0809.4266] [INSPIRE].
K. Glampedakis and N. Andersson, Late time dynamics of rapidly rotating black holes, Phys. Rev. D 64 (2001) 104021 [gr-qc/0103054] [INSPIRE].
M. Casals, S.E. Gralla and P. Zimmerman, Horizon Instability of Extremal Kerr Black Holes: Nonaxisymmetric Modes and Enhanced Growth Rate, Phys. Rev. D 94 (2016) 064003 [arXiv:1606.08505] [INSPIRE].
R.A. Konoplya and A. Zhidenko, Massive charged scalar field in the Kerr-Newman background I: quasinormal modes, late-time tails and stability, Phys. Rev. D 88 (2013) 024054 [arXiv:1307.1812] [INSPIRE].
S.E. Gralla and P. Zimmerman, Critical Exponents of Extremal Kerr Perturbations, Class. Quant. Grav. 35 (2018) 095002 [arXiv:1711.00855] [INSPIRE].
T. Faulkner, H. Liu, J. McGreevy and D. Vegh, Emergent quantum criticality, Fermi surfaces and AdS 2, Phys. Rev. D 83 (2011) 125002 [arXiv:0907.2694] [INSPIRE].
G. Compère, K. Fransen, T. Hertog and J. Long, Gravitational waves from plunges into Gargantua, Class. Quant. Grav. 35 (2018) 104002 [arXiv:1712.07130] [INSPIRE].
M. Casals and P. Zimmerman, Perturbations of Extremal Kerr Spacetime: Analytic Framework and Late-time Tails, arXiv:1801.05830 [INSPIRE].
K. Prabhu, The First Law of Black Hole Mechanics for Fields with Internal Gauge Freedom, Class. Quant. Grav. 34 (2017) 035011 [arXiv:1511.00388] [INSPIRE].
P. Zimmerman, Horizon instability of extremal Reissner-Nordström black holes to charged perturbations, Phys. Rev. D 95 (2017) 124032 [arXiv:1612.03172] [INSPIRE].
S.E. Gralla, A. Zimmerman and P. Zimmerman, Transient Instability of Rapidly Rotating Black Holes, Phys. Rev. D 94 (2016) 084017 [arXiv:1608.04739] [INSPIRE].
L.M. Burko and G. Khanna, Linearized Stability of Extreme Black Holes, Phys. Rev. D 97 (2018) 061502 [arXiv:1709.10155] [INSPIRE].
S. Hadar and H.S. Reall, Is there a breakdown of effective field theory at the horizon of an extremal black hole?, JHEP 12 (2017) 062 [arXiv:1709.09668] [INSPIRE].
D. Basu, Introduction to Classical and Modern Analysis and Their Application to Group Representation Theory, World Scientific, (2011).
A.O. Barut and C. Fronsdal, On Non-Compact Groups. II. Representations of the 2 + 1 Lorentz Group, Proc. Roy. Soc. Lond. A 287 (1965) 532.
P. Breitenlohner and D.Z. Freedman, Positive Energy in anti-de Sitter Backgrounds and Gauged Extended Supergravity, Phys. Lett. B 115 (1982) 197 [INSPIRE].
P. Breitenlohner and D.Z. Freedman, Stability in Gauged Extended Supergravity, Annals Phys. 144 (1982) 249 [INSPIRE].
A. Ishibashi and R.M. Wald, Dynamics in nonglobally hyperbolic static space-times. 3. Anti-de Sitter space-time, Class. Quant. Grav. 21 (2004) 2981 [hep-th/0402184] [INSPIRE].
M. Kleban, M. Porrati and R. Rabadán, Stability in asymptotically AdS spaces, JHEP 08 (2005) 016 [hep-th/0409242] [INSPIRE].
G. Holzegel, Well-posedness for the massive wave equation on asymptotically anti-de Sitter spacetimes, arXiv:1103.0710 [INSPIRE].
C.M. Warnick, The massive wave equation in asymptotically AdS spacetimes, Commun. Math. Phys. 321 (2013) 85 [arXiv:1202.3445] [INSPIRE].
DLMF, NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/ Release 1.0.5 of 2012-10-01.
H. Bateman and A. Erdélyi, Higher transcendental functions, Calif. Inst. Technol, Bateman Manuscr. Project McGraw-Hill, New York, NY, (1955), https://cds.cern.ch/record/100233.
B.M. Project, H. Bateman, A. Erdélyi and U.S.O. of Naval Research, Tables of Integral Transforms: Based, in Part, on Notes Left by Harry Bateman, v. 2, McGraw-Hill, (1954).
L. Slater, Confluent hypergeometric functions, Cambridge University Press, (1960).
D.T. Son and A.O. Starinets, Minkowski space correlators in AdS/CFT correspondence: Recipe and applications, JHEP 09 (2002) 042 [hep-th/0205051] [INSPIRE].
H.S. Reall, Higher dimensional black holes and supersymmetry, Phys. Rev. D 68 (2003) 024024 [Erratum ibid. D 70 (2004) 089902] [hep-th/0211290] [INSPIRE].
D. Astefanesei, K. Goldstein, R.P. Jena, A. Sen and S.P. Trivedi, Rotating attractors, JHEP 10 (2006) 058 [hep-th/0606244] [INSPIRE].
H.K. Kunduri and J. Lucietti, Classification of near-horizon geometries of extremal black holes, Living Rev. Rel. 16 (2013) 8 [arXiv:1306.2517] [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].
D.D.K. Chow, M. Cvetič, H. Lü and C.N. Pope, Extremal Black Hole/CFT Correspondence in (Gauged) Supergravities, Phys. Rev. D 79 (2009) 084018 [arXiv:0812.2918] [INSPIRE].
S. Hollands and A. Ishibashi, Instabilities of extremal rotating black holes in higher dimensions, Commun. Math. Phys. 339 (2015) 949 [arXiv:1408.0801] [INSPIRE].
S. Hollands, A. Ishibashi and R.M. Wald, A higher dimensional stationary rotating black hole must be axisymmetric, Commun. Math. Phys. 271 (2007) 699 [gr-qc/0605106] [INSPIRE].
J. Ren, Analytic quantum critical points from holography, arXiv:1210.2722 [INSPIRE].
S.A. Hartnoll, A. Lucas and S. Sachdev, Holographic quantum matter, arXiv:1612.07324 [INSPIRE].
J.M. Cohen and L.S. Kegeles, Electromagnetic fields in curved spaces - a constructive procedure, Phys. Rev. D 10 (1974) 1070 [INSPIRE].
P.L. Chrzanowski, Vector Potential and Metric Perturbations of a Rotating Black Hole, Phys. Rev. D 11 (1975) 2042 [INSPIRE].
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].
J.M. Stewart, Hertz-Bromwich-Debye-Whittaker-Penrose Potentials in General Relativity, Proc. Roy. Soc. Lond. A 367 (1979) 527 [INSPIRE].
M. Godazgar, The perturbation theory of higher dimensional spacetimes a la Teukolsky, Class. Quant. Grav. 29 (2012) 055008 [arXiv:1110.5779] [INSPIRE].
R.M. Wald, On perturbations of a Kerr black hole, J. Math. Phys. 14 (1973) 1453.
B. Carter, Hamilton-Jacobi and Schrödinger separable solutions of Einstein’s equations, Commun. Math. Phys. 10 (1968) 280 [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].
T. Hartman, K. Murata, T. Nishioka and A. Strominger, CFT Duals for Extreme Black Holes, JHEP 04 (2009) 019 [arXiv:0811.4393] [INSPIRE].
G. Compère, The Kerr/CFT correspondence and its extensions, Living Rev. Rel. 15 (2012) 11 [arXiv:1203.3561] [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].
L.J. Romans, Supersymmetric, cold and lukewarm black holes in cosmological Einstein-Maxwell theory, Nucl. Phys. B 383 (1992) 395 [hep-th/9203018] [INSPIRE].
A. Chamblin, R. Emparan, C.V. Johnson and R.C. Myers, Charged AdS black holes and catastrophic holography, Phys. Rev. D 60 (1999) 064018 [hep-th/9902170] [INSPIRE].
S.E. Gralla, A. Ravishankar and P. Zimmerman, in preparation.
K. Murata, Conformal weights in the Kerr/CFT correspondence, JHEP 05 (2011) 117 [arXiv:1103.5635] [INSPIRE].
F.W. Olver, D.W. Lozier, R.F. Boisvert and C.W. Clark, NIST Handbook of Mathematical Functions, 1st ed., Cambridge University Press, New York, NY, U.S.A., (2010).
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Gralla, S.E., Zimmerman, P. Scaling and universality in extremal black hole perturbations. J. High Energ. Phys. 2018, 61 (2018). https://doi.org/10.1007/JHEP06(2018)061
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DOI: https://doi.org/10.1007/JHEP06(2018)061