Annales Henri Poincaré

, Volume 13, Issue 5, pp 1101–1166 | Cite as

Asymptotic Distribution of Quasi-Normal Modes for Kerr–de Sitter Black Holes

Article

Abstract

We establish a Bohr–Sommerfeld type condition for quasi-normal modes of a slowly rotating Kerr–de Sitter black hole, providing their full asymptotic description in any strip of fixed width. In particular, we observe a Zeeman-like splitting of the high multiplicity modes at a = 0 (Schwarzschild–de Sitter), once spherical symmetry is broken. The numerical results presented in Appendix B show that the asymptotics are in fact accurate at very low energies and agree with the numerical results established by other methods in the physics literature. We also prove that solutions of the wave equation can be asymptotically expanded in terms of quasi-normal modes; this confirms the validity of the interpretation of their real parts as frequencies of oscillations, and imaginary parts as decay rates of gravitational waves.

References

  1. 1.
    Abbott B.P. et al.: The LIGO scientic collaboration, search for gravitational wave ringdowns from perturbed black holes in LIGO S4 data. Phys. Rev. D 80, 062001 (2009)MathSciNetADSCrossRefGoogle Scholar
  2. 2.
    Andersson, L., Blue, P.: Hidden symmetries and decay for the wave equation on the Kerr spacetime. Preprint, arXiv:0908.2265Google Scholar
  3. 3.
    Bachelot A.: Gravitational scattering of electromagnetic field by Schwarzschild black hole. Ann. Inst. H. Poincaré Phys. Théor. 54, 261–320 (1991)MathSciNetADSMATHGoogle Scholar
  4. 4.
    Bachelot A.Scattering of electromagnetic field by de Sitter–Schwarzschild black hole. Non-linear hyperbolic equations and field theory. Pitman Res. Notes Math. Ser. 253, 23–35Google Scholar
  5. 5.
    Bachelot A., Motet-Bachelot A.: Les résonances d’un trou noir de Schwarzschild. Ann. Inst. H. Poincaré Phys. Théor. 59, 3–68 (1993)MathSciNetMATHGoogle Scholar
  6. 6.
    Berti E., Cardoso V., Starinets A.: Quasinormal modes of black holes and black branes. Class. Quant. Grav. 26, 163001 (2009)MathSciNetADSCrossRefGoogle Scholar
  7. 7.
    Berti E., Cardoso V., Will C.M.: On gravitational-wave spectroscopy of massive black holes with the space interferometer LISA. Phys. Rev. D 73, 064030 (2006)MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    Berti E., Kokkotas K.: Quasinormal modes of Kerr–Newman black holes: coupling of electromagnetic and gravitational perturbations. Phys. Rev. D 71, 124008 (2005)MathSciNetADSCrossRefGoogle Scholar
  9. 9.
    Blue P., Sterbenz J.: Uniform decay of local energy and the semi-linear wave equation on Schwarzschild space. Comm. Math. Phys. 268, 481–504 (2006)MathSciNetADSMATHCrossRefGoogle Scholar
  10. 10.
    Bony J.-F., Häfner D.: Decay and non-decay of the local energy for the wave equation on the de Sitter-Schwarzschild metric. Comm. Math. Phys. 282, 697–719 (2008)MathSciNetADSMATHCrossRefGoogle Scholar
  11. 11.
    Chandrasekhar S.: The Mathematical Theory of Black Holes. Oxford Classic Texts in the Physical Sciences. Oxford University Press, Oxford (2000)Google Scholar
  12. 12.
    Charbonnel A.-M.: Spectre conjoint d’opérateurs pseudodifférentiels qui commutent. Ann. Fac. Sci. Toulouse Math. 5, 109–147 (1983)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Christiansen T., Zworski M.: Resonance wave expansions: two hyperbolic examples. Comm. Math. Phys. 212, 323–336 (2000)MathSciNetADSMATHCrossRefGoogle Scholar
  14. 14.
    Colinde Verdière Y.: Bohr–Sommerfeld rules to all orders. Ann. Henri Poincaré 6, 925–936 (2005)ADSCrossRefGoogle Scholar
  15. 15.
    Colin de Verdière, Y., Guillemin, V.: A semi-classical inverse problem I: Taylor expansions. Preprint, arXiv:0802.1605Google Scholar
  16. 16.
    Colinde Verdière Y., Parisse B.: Équilibre instable en régime semi-classique: I—Concentration microlocale. Comm. Partial Differ. Equ. 19, 1535–1563 (1994)CrossRefGoogle Scholar
  17. 17.
    Dafermos, M., Rodnianski, I.: The wave equation on Schwarzschild–de Sitter spacetimes. Preprint, arXiv:0709.2766Google Scholar
  18. 18.
    Dafermos, M., Rodnianski, I.: Lectures on black holes and linear waves. Preprint, arXiv:0811.0354v1Google Scholar
  19. 19.
    Dafermos, M., Rodnianski, I.: Decay for solutions of the wave equation on Kerr exterior space-times I–II: the cases of |a| ≪ M or axisymmetry. Preprint, arXiv:1010.5132Google Scholar
  20. 20.
    Datchev, K., Vasy, A.: Gluing semiclassical resolvent estimates via propagation of singularities. Preprint, arXiv:1008.3964Google Scholar
  21. 21.
    Dimassi M., Sjöstrand J.: Spectral Asymptotics in the Semi-classical Limit. Cambridge University Press, Cambridge (1999)MATHCrossRefGoogle Scholar
  22. 22.
    Donninger, R., Schlag, W., Soffer, A.: A proof of Price’s Law on Schwarzschild black hole manifolds for all angular momenta. Preprint, arXiv:0908.4292Google Scholar
  23. 23.
    Donninger, R., Schlag, W., Soffer, A.: On pointwise decay of linear waves on a Schwarzschild black hole background. Preprint, arXiv:0911.3179Google Scholar
  24. 24.
    Duistermaat J.J.: On global action-angle coordinates. Comm. Pure Appl. Math. 33, 687–706 (1980)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Dunford N., Schwarz J.T.: Linear Operators, Part I: General Theory. Interscience, New York (1958)Google Scholar
  26. 26.
    Dyatlov S.: Quasi-normal modes and exponential energy decay for the Kerr–de Sitter black hole. Comm. Math. Phys. 306, 119–163 (2011)MathSciNetADSMATHCrossRefGoogle Scholar
  27. 27.
    Dyatlov, S.: Exponential energy decay for Kerr–de Sitter black holes beyond event horizons. Math. Res. Lett. (to appear), arXiv:1010.5201Google Scholar
  28. 28.
    Finster F., Kamran N., Smoller J., Yau S.-T.: Decay of solutions of the wave equation in the Kerr geometry. Comm. Math. Phys. 264, 465–503 (2006)MathSciNetADSMATHCrossRefGoogle Scholar
  29. 29.
    Finster F., Kamran N., Smoller J., Yau S.-T.: Erratum: decay of solutions of the wave equation in the Kerr geometry. Comm. Math. Phys. 280, 563–573 (2008)MathSciNetADSMATHCrossRefGoogle Scholar
  30. 30.
    Guillemin, V., Sternberg, S.: Semi-classical analysis. Lecture Notes (version of January 13, 2010), http://www-math.mit.edu/~vwg/semiclassGuilleminSternberg.pdf
  31. 31.
    Hitrik M., Sjöstrand J.: Non-selfadjoint perturbations of selfadjoint operators in two dimensions. I. Ann. Henri Poincaré 5, 1–73 (2004)MATHCrossRefGoogle Scholar
  32. 32.
    Hitrik M., Sjöstrand J., SanVũ Ngọc : Diophantine tori and spectral asymptotics for nonselfadjoint operators. Am. J. Math. 129, 105–182 (2007)MATHGoogle Scholar
  33. 33.
    Horowitz G.T., Hubeny V.E.: Quasinormal modes of AdS black holes and the approach to thermal equilibrium. Phys. Rev. D 62, 024027 (2000)MathSciNetADSCrossRefGoogle Scholar
  34. 34.
    Hörmander L.: The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis. Springer, Berlin (1990)MATHGoogle Scholar
  35. 35.
    Kokkotas, K.D., Schmidt, B.G.: Quasi-normal modes of stars and black holes. Living Rev. Relativ. 2 (1999), http://www.livingreviews.org/lrr-1999-2
  36. 36.
    Konoplya R.A., Zhidenko A.: High overtones of Schwarzschild-de Sitter quasinormal spectrum. JHEP 0406, 037 (2004)MathSciNetADSCrossRefGoogle Scholar
  37. 37.
    Konoplya R.A., Zhidenko A.: Decay of a charged scalar and Dirac fields in the Kerr-Newman-de Sitter background. Phys. Rev. D 76, 084018 (2007)MathSciNetADSCrossRefGoogle Scholar
  38. 38.
    Melrose, R., Sá Barreto, A., Vasy, A.: Asymptotics of solutions of the wave equation on de Sitter–Schwarzschild space. Preprint, arXiv:0811.2229Google Scholar
  39. 39.
    Pravica D.: Top resonances of a black hole. R. Soc. Lond. Proc. Ser. A 455, 3003–3018 (1999)MathSciNetADSMATHCrossRefGoogle Scholar
  40. 40.
    Ramond T.: Semiclassical study of quantum scattering on the line. Comm. Math. Phys. 177, 221–254 (1996)MathSciNetADSMATHCrossRefGoogle Scholar
  41. 41.
    Sá~Barreto A., Zworski M.: Distribution of resonances for spherical black holes. Math. Res. Lett. 4, 103–121 (1997)MathSciNetMATHGoogle Scholar
  42. 42.
    Sjöstrand, J.: Semiclassical resonances generated by nondegenerate critical points. Pseudodifferential operators, Lecture Notes in Mathematics, vol. 1256, pp. 402–429. Springer, Berlin (1987)Google Scholar
  43. 43.
    Sjöstrand J.: Semi-excited states in nondegenerate potential wells. Asymp. Anal. 6, 29–43 (1992)MATHGoogle Scholar
  44. 44.
    Sjöstrand J., Zworski M.: Quantum monodromy and semi-classical trace formulae. J. Math. Pures Appl. 81, 1–33 (2002)MathSciNetMATHGoogle Scholar
  45. 45.
    Tang S.-H., Zworski M.: Resonance expansions of scattered waves. Comm. Pure Appl. Math. 53, 1305–1334 (2000)MathSciNetMATHCrossRefGoogle Scholar
  46. 46.
    Tataru, D.: Local decay of waves on asymptotically flat stationary space-times. Preprint, arXiv:0910.5290Google Scholar
  47. 47.
    Tataru, D., Tohaneanu, M.: Local energy estimate on Kerr black hole backgrounds. Preprint, arXiv:0810.5766Google Scholar
  48. 48.
    Taylor M.: Partial Differential Equations, I. Basic Theory. Springer, Berlin (1996)Google Scholar
  49. 49.
    Tohaneanu, M.: Strichartz estimates on Kerr black hole backgrounds. Preprint, arXiv:0910.1545Google Scholar
  50. 50.
    Vasy, A.: Microlocal analysis of asymptotically hyperbolic and Kerr–de Sitter spaces. Preprint, arXiv:1012.4391Google Scholar
  51. 51.
    Vũ Ngọc S.: Systèmes intégrables semi-classiques: du local au global. Société Mathématique de France, France (2006)Google Scholar
  52. 52.
    Wunsch, J., Zworski, M.: Resolvent estimates for normally hyperbolic trapped sets. Preprint, arXiv:1003.4640Google Scholar
  53. 53.
    Yoshida S., Uchikata N., Futamase T.: Quasinormal modes of Kerr–de Sitter black holes. Phys. Rev. D 81, 044005 (2010)MathSciNetADSCrossRefGoogle Scholar
  54. 54.
    Zworski, M.: Semiclassical analysis. Graduate Studies in Mathematics. AMS (2012), http://math.berkeley.edu/~zworski/semiclassical.pdf

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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