no. 3
A global definition of quasinormal modes for Kerr–AdS black holes
[Une définition globale des modes quasinormaux pour les trous noirs Kerr–AdS]
Gannot, Oran1
1 Department of Mathematics, Northwestern University 2033 Sheridan Rd. Evanston, IL 60202 (USA)
Annales de l'Institut Fourier, Tome 68 (2018) no. 3, pp. 1125-1167.

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.

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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : 10.5802/aif.3186
Classification : 35P25, 58J50, 37Q75, 35B34, 35L05
Keywords: Kerr–AdS black holes, quasinormal modes, scattering theory
Mot clés : Kerr-AdS trous noirs, modes quasinormaux, théorie de la diffusion

Gannot, Oran 1

1 Department of Mathematics, Northwestern University 2033 Sheridan Rd. Evanston, IL 60202 (USA)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Gannot, Oran. A global definition of quasinormal modes for Kerr–AdS black holes. Annales de l'Institut Fourier, Tome 68 (2018) no. 3, pp. 1125-1167. doi : 10.5802/aif.3186. https://aif.centre-mersenne.org/articles/10.5802/aif.3186/

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

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

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

[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., Volume 113 (2014) no. 7, 071601 (Article ID 071601) | DOI

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

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

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

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

[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., Volume 282 (2008) no. 3, pp. 697-719 | DOI | Zbl

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

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

[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, Volume 91 (2015) no. 6, 064026 (Article ID 064026) | DOI

[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., Volume 2014 (2014), 183, 71 pages (Article ID 183, 71 p.) | DOI | Zbl

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

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

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

[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., Volume 29 (2012), 235019, 24 pages (Article ID 235019, 24 p.) | DOI | Zbl

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

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

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

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

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

[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., Volume 17 (1995) no. 2, pp. 125-184 | DOI | Zbl

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

[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, Volume 18 (2017) no. 8, pp. 1-32 | DOI | Zbl

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

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

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

[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., Volume 66 (2013) no. 11, pp. 1751-1802 | DOI | Zbl

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

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

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

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

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

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

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

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

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

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

[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, Volume 4 (1997) no. 1, pp. 103-121 | DOI | Zbl

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

[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., Volume 223 (2010) no. 1, pp. 49-97 | DOI | Zbl

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

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

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

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

[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, Graduate Studies in Mathematics, 138, American Mathematical Society, 2012, xii+431 pages | Zbl

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

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