A global definition of quasinormal modes for Kerr–AdS black holes
Annales de l'Institut Fourier, Volume 68 (2018) no. 3, p. 1125-1167
The quasinormal frequencies of massive scalar fields on Kerr–AdS black holes are identified with poles of a certain meromorphic family of operators, once boundary conditions are specified at the conformal boundary. Consequently, the quasinormal frequencies form a discrete subset of the complex plane and the corresponding poles are of finite rank. This result holds for a broad class of elliptic boundary conditions, with no restrictions on the rotation speed of the black hole.
Les fréquences quasinormales des champs scalaires massifs sur les trous noirs Kerr–AdS sont identifiées avec les pôles d’une certaine famille d’opérateurs méromorphes, une fois que les conditions limites sont spécifiées à la limite conforme. Par conséquent, les fréquences quasinormales forment un sous-ensemble discret du plan complexe et les pôles correspondants sont de rang fini. Ce résultat réside dans une large classe de conditions aux limites elliptiques, sans aucune restriction sur la vitesse de rotation du trou noir.
Received : 2016-01-01
Revised : 2017-06-01
Accepted : 2017-08-07
Published online : 2018-05-04
DOI : https://doi.org/10.5802/aif.3186
Classification:  35P25,  58J50,  37Q75,  35B34,  35L05
Keywords: Kerr–AdS black holes, quasinormal modes, scattering theory
@article{AIF_2018__68_3_1125_0,
     author = {Gannot, Oran},
     title = {A global definition of quasinormal modes for Kerr--AdS black holes},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {68},
     number = {3},
     year = {2018},
     pages = {1125-1167},
     doi = {10.5802/aif.3186},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2018__68_3_1125_0}
}
A global definition of quasinormal modes for Kerr–AdS black holes. Annales de l'Institut Fourier, Volume 68 (2018) no. 3, pp. 1125-1167. doi : 10.5802/aif.3186. https://aif.centre-mersenne.org/item/AIF_2018__68_3_1125_0/

[1] Avis, S. J.; Isham, Christopher J.; Storey, D. Quantum field theory in anti-de Sitter space-time, Phys. Rev. D, Tome 18 (1978), pp. 3565-3576 | Article

[2] Bachelot, Alain Gravitational scattering of electromagnetic field by Schwarzschild black-hole, Ann. Inst. Henri Poincaré, Phys. Théor., Tome 54 (1991) no. 3, pp. 261-320 | Zbl 0743.53037

[3] Bachelot, Alain; Motet-Bachelot, Agnès Les résonances d’un trou noir de Schwarzschild, Ann. Inst. Henri Poincaré, Phys. Théor., Tome 59 (1993) no. 1, pp. 3-68 | Zbl 0793.53094

[4] Balasubramanian, V.; Buchel, Alex; Green, Stephen R.; Lehner, Luis; Liebling, Steven L. Holographic Thermalization, Stability of Anti–de Sitter Space, and the Fermi-Pasta-Ulam Paradox, Phys. Rev. Lett., Tome 113 (2014) no. 7, 071601 (Article ID 071601) | Article

[5] Berkooz, Micha; Sever, Amit; Shomer, Assaf ‘Double-trace’ deformations, boundary conditions and spacetime singularities, J. High Energy Phys., Tome 2002 (2002) no. 05, 034 (Article ID 034) | Article

[6] Bizoń, Piotr Is AdS stable?, Gen. Relativ. Gravitation, Tome 46 (2014) no. 5, 1724, 14 pages (Article ID 1724, 14 p.) | Article | Zbl 1291.83003

[7] Bizoń, Piotr; Maliborski, Maciej; Rostworowski, Andrzej Resonant Dynamics and the Instability of Anti–de Sitter Spacetime, Phys. Rev. Lett., Tome 115 (2015) no. 8, 081103 (Article ID 08103) | Article

[8] Bizoń, Piotr; Rostworowski, Andrzej Weakly Turbulent Instability of Anti de Sitter Spacetime, Phys. Rev. Lett., Tome 107 (2011), 031102 (Article ID 031102) | Article | Zbl 10.1103/PhysRevLett.107.031102

[9] Bony, Jean-François; Häfner, Dietrich Decay and Non-Decay of the Local Energy for the Wave Equation on the De Sitter-Schwarzschild Metric, Commun. Math. Phys., Tome 282 (2008) no. 3, pp. 697-719 | Article | Zbl 159.35007

[10] Breitenlohner, Peter; Freedman, Daniel Z. Positive Energy in anti-de Sitter Backgrounds and Gauged Extended Supergravity, Phys. Lett., Tome 115 (1982), pp. 197-201 | Article

[11] Breitenlohner, Peter; Freedman, Daniel Z. Stability in gauged extended supergravity, Ann. Phys., Tome 144 (1982) no. 2, pp. 249-281 | Article | Zbl 0606.53044

[12] Buchel, Alex; Green, Stephen R.; Lehner, Luis; Liebling, Steven L. Conserved quantities and dual turbulent cascades in anti–de Sitter spacetime, Phys. Rev. D, Tome 91 (2015) no. 6, 064026 (Article ID 064026) | Article

[13] Cardoso, Vitor; Dias, Óscar J. C.; Hartnett, Gavin S.; Lehner, Luis; Santos, Jorge E. Holographic thermalization, quasinormal modes and superradiance in Kerr–AdS, J. High Energy Phys., Tome 2014 (2014), 183, 71 pages (Article ID 183, 71 p.) | Article | Zbl 1333.86063

[14] Carter, Brandon Hamilton-Jacobi and Schrödinger separable solutions of Einstein’s equations, Commun. Math. Phys., Tome 10 (1968) no. 4, pp. 280-310 | Article | Zbl 0162.59302

[15] Craps, Ben; Evnin, Oleg; Vanhoof, Joris Renormalization group, secular term resummation and AdS (in)stability, J. High Energy Phys., Tome 2014 (2014) no. 10, 048, 30 pages (Article ID 048, 30 p.) | Article | Zbl 1333.83045

[16] Craps, Ben; Evnin, Oleg; Vanhoof, Joris Renormalization, averaging, conservation laws and AdS (in)stability, J. High Energy Phys., Tome 2015 (2015) no. 01, 108 (Article ID 108) | Article

[17] Dias, Óscar J. C.; Horowitz, Gary T.; Marolf, Don; Santos, Jorge E. On the Nonlinear Stability of Asymptotically Anti-de Sitter Solutions, Class. Quant. Grav., Tome 29 (2012), 235019, 24 pages (Article ID 235019, 24 p.) | Article | Zbl 1258.873022

[18] Dias, Óscar J. C.; Horowitz, Gary T.; Santos, Jorge E. Gravitational turbulent instability of anti-de Sitter space, Class. Quant. Grav., Tome 29 (2012) no. 19, 194002, 7 pages (Article ID 194002, 7 p.) | Article | Zbl 1254.83030

[19] Dias, Óscar J. C.; Santos, Jorge E. Boundary conditions for Kerr–AdS perturbations, J. High Energy Phys., Tome 2013 (2013) no. 10, 156, 37 pages (Article ID 156, 37 p.) | Article | Zbl 1342.83148

[20] Dimassi, Mouez; Sjöstrand, Johannes Spectral Asymptotics in the Semi-Classical Limit, Cambridge University Press, London Mathematical Society Lecture Note Series, Tome 268 (1999), xi+227 pages | Zbl 0926.35002

[21] Dyatlov, Semyon Quasi-Normal Modes and Exponential Energy Decay for the Kerr-de Sitter Black Hole, Commun. Math. Phys., Tome 306 (2011) no. 1, pp. 119-163 | Article | Zbl 1223.83029

[22] Dyatlov, Semyon Asymptotic Distribution of Quasi-Normal Modes for Kerr-de Sitter Black Holes, Ann. Henri Poincaré, Tome 13 (2012) no. 5, pp. 1101-1166 | Article | Zbl 1246.83111

[23] Dyatlov, Semyon; Zworski, Maciej Mathematical theory of scattering resonances (http://math.mit.edu/~dyatlov/res/res_20180326.pdf )

[24] Friedrich, Helmut Einstein equations and conformal structure: existence of anti-de Sitter-type space-times, J. Geom. Phys., Tome 17 (1995) no. 2, pp. 125-184 | Article | Zbl 0840.53055

[25] Gannot, Oran Quasinormal Modes for Schwarzschild-AdS Black Holes: Exponential Convergence to the Real Axis, Commun. Math. Phys., Tome 330 (2014) no. 2, pp. 771-799 | Article | Zbl 1295.85001

[26] Gannot, Oran Elliptic boundary value problems for Bessel operators, with applications to anti-de Sitter spacetimes (2015) (https://arxiv.org/abs/1507.02794 )

[27] Gannot, Oran Existence of Quasinormal Modes for Kerr–AdS Black Holes, Ann. Henri Poincaré, Tome 18 (2017) no. 8, pp. 1-32 | Article | Zbl 1375.83025

[28] Giammatteo, M.; Moss, Ian G. Gravitational quasinormal modes for Kerr anti-de Sitter black holes, Class. Quant. Grav., Tome 22 (2005) no. 9, pp. 1803-1824 | Article | Zbl 1068.83009

[29] Hawking, S. W.; Hunter, C. J.; Taylor, Michael E. Rotation and the AdS/CFT correspondence, Phys. Rev. D, Tome 59 (1999), 064005 (Article ID 064005) | Article

[30] Hintz, Peter; Vasy, András Semilinear wave equations on asymptotically de Sitter, Kerr-de Sitter and Minkowski spacetimes, Anal. PDE, Tome 8 (2013) no. 8, pp. 1807-1890 | Article | Zbl 1336.35244

[31] Hintz, Peter; Vasy, András Asymptotics for the wave equation on differential forms on Kerr-de Sitter space (2015) (https://arxiv.org/abs/1502.03179 )

[32] Holzegel, Gustav; Luk, Jonathan; Smulevici, Jacques; Warnick, Claude M. Asymptotic properties of linear field equations in anti-de Sitter space (2015) (https://arxiv.org/abs/1502.04965 )

[33] Holzegel, Gustav; Smulevici, Jacques Decay Properties of Klein–Gordon Fields on Kerr–AdS Spacetimes, Commun. Pure Appl. Math., Tome 66 (2013) no. 11, pp. 1751-1802 | Article | Zbl 1277.83023

[34] Holzegel, Gustav; Smulevici, Jacques Stability of Schwarzschild–AdS for the spherically symmetric Einstein-Klein-Gordon system, Commun. Math. Phys., Tome 317 (2013) no. 1, pp. 205-251 | Article | Zbl 1259.83012

[35] Holzegel, Gustav; Smulevici, Jacques Quasimodes and a lower bound on the uniform energy decay rate for Kerr–AdS spacetimes, Anal. PDE, Tome 7 (2014) no. 5, pp. 1057-1090 | Article | Zbl 1300.83030

[36] Holzegel, Gustav; Warnick, Claude M. Boundedness and growth for the massive wave equation on asymptotically anti-de Sitter black holes, J. Funct. Anal., Tome 266 (2014) no. 4, pp. 2436-2485 | Article | Zbl 1292.83033

[37] Hörmander, Lars The Analysis of Linear Partial Differential Operators III: Pseudo-Differential Operators, Springer, Grundlehren der Mathematischen Wissenschaften, Tome 274 (1985), viii+525 pages | Zbl 0601.35001

[38] Hörmander, Lars The Analysis of Linear Partial Differential Operators IV: Fourier Integral Operators, Springer, Classics in Mathematics (2009), vii+352 pages | Zbl 1178.35003

[39] Horowitz, Gary T.; Hubeny, Veronika E. Quasinormal modes of AdS black holes and the approach to thermal equilibrium, Phys. Rev. D, Tome 62 (2000), 024027, 11 pages (Article ID 024027, 11 p.) | Article

[40] Ishibashi, Akihiro; Wald, Robert M. Dynamics in nonglobally hyperbolic static space-times III: Anti-de Sitter space-time, Class. Quant. Grav., Tome 21 (2004) no. 12, pp. 2981-3014 | Article

[41] Konoplya, R. A.; Zhidenko, Alexander Quasinormal modes of black holes: From astrophysics to string theory, Rev. Mod. Phys., Tome 83 (2011), pp. 793-836 | Article

[42] Melrose, Richard B. Spectral and scattering theory for the Laplacian on asymptotically Euclidean spaces, Spectral and scattering theory, Marcel Dekker AG (Lecture Notes in Pure and Applied Mathematics) (1994), p. 85-85 | Zbl 0837.35107

[43] Boutet De Monvel, Louis Boundary problems for pseudo-differential operators, Acta Math., Tome 126 (1971) no. 1, pp. 11-51 | Article | Zbl 0206.39401

[44] Olver, Frank William John Introduction to Asymptotics and Special Functions, Elsevier (2014)

[45] Sá Barreto, Antônio; Zworski, Maciej Distribution of resonances for spherical black holes, Math. Res. Lett, Tome 4 (1997) no. 1, pp. 103-121 | Article | Zbl 0883.35120

[46] Shubin, Mikhail Aleksandrovich Pseudodifferential Operators and Spectral Theory, Springer (2001), xii+288 pages | Zbl 0980.35180

[47] Taylor, Michael E. Partial Differential Equations I: Basic Theory, Springer (1996)

[48] Vasy, András The wave equation on asymptotically de Sitter-like spaces, Adv. Math., Tome 223 (2010) no. 1, pp. 49-97 | Article | Zbl 1192.35064

[49] Vasy, András Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces, Invent. Math., Tome 194 (2013) no. 2, pp. 381-513 | Article | Zbl 1315.35015

[50] Warnick, Claude M. The Massive Wave Equation in Asymptotically AdS Spacetimes, Commun. Math. Phys., Tome 321 (2013) no. 1, pp. 85-111 | Article | Zbl 1271.83021

[51] Warnick, Claude M. On Quasinormal Modes of Asymptotically Anti-de Sitter Black Holes, Commun. Math. Phys., Tome 333 (2015) no. 2, pp. 959-1035 | Article | Zbl 1308.83107

[52] Witten, Edward Anti-de Sitter space and holography, Adv. Theor. Math. Phys., Tome 2 (1998) no. 2, pp. 253-291 | Article | Zbl 0914.53048

[53] Witten, Edward Multi-trace operators, boundary conditions, and AdS/CFT correspondence (2001) (https://arxiv.org/abs/hep-th/0112258 )

[54] Zworski, Maciej Semiclassical Analysis, American Mathematical Society, Graduate Studies in Mathematics, Tome 138 (2012), xii+431 pages | Zbl 1252.58001

[55] Zworski, Maciej Resonances for asymptotically hyperbolic manifolds: Vasy’s method revisited, J. Spectr. Theory, Tome 6 (2016) no. 4, pp. 1087-1114 | Article | Zbl 1365.58012