Kerr-de Sitter quasinormal modes via accessory parameter expansion

  • Fábio NovaesEmail author
  • Cássio I. S. Marinho
  • Máté Lencsés
  • Marc Casals
Open Access
Regular Article - Theoretical Physics


Quasinormal modes are characteristic oscillatory modes that control the relaxation of a perturbed physical system back to its equilibrium state. In this work, we calculate QNM frequencies and angular eigenvalues of Kerr-de Sitter black holes using a novel method based on conformal field theory. The spin-field perturbation equations of this background spacetime essentially reduce to two Heun’s equations, one for the radial part and one for the angular part. We use the accessory parameter expansion of Heun’s equation, obtained via the isomonodromic τ -function, in order to find analytic expansions for the QNM frequencies and angular eigenvalues. The expansion for the frequencies is given as a double series in the rotation parameter a and the extremality parameter ϵ = (rCr+)/L, where L is the de Sitter radius and rC and r+ are the radii of, respectively, the cosmological and event horizons. Specifically, we give the frequency expansion up to order ϵ2 for general a, and up to order ϵ3 with the coefficients expanded up to (a/L)3. Similarly, the expansion for the angular eigenvalues is given as a series up to ()3 with coefficients expanded for small a/L. We verify the new expansion for the frequencies via a numerical analysis and that the expansion for the angular eigenvalues agrees with results in the literature.


Black Holes Classical Theories of Gravity Conformal Field Theory 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


  1. [1]
    B.F. Whiting, Mode Stability of the Kerr Black Hole, J. Math. Phys. 30 (1989) 1301 [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    LIGO Scientific and Virgo collaborations, Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett. 116 (2016) 061102 [arXiv:1602.03837] [INSPIRE].
  3. [3]
    G.T. Horowitz and V.E. Hubeny, Quasinormal modes of AdS black holes and the approach to thermal equilibrium, Phys. Rev. D 62 (2000) 024027 [hep-th/9909056] [INSPIRE].
  4. [4]
    K.D. Kokkotas and B.G. Schmidt, Quasinormal modes of stars and black holes, Living Rev. Rel. 2 (1999) 2 [gr-qc/9909058] [INSPIRE].
  5. [5]
    E. Berti, V. Cardoso and A.O. Starinets, Quasinormal modes of black holes and black branes, Class. Quant. Grav. 26 (2009) 163001 [arXiv:0905.2975] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    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].
  7. [7]
    O.J.C. Dias, J.E. Santos and M. Stein, Kerr-AdS and its Near-horizon Geometry: Perturbations and the Kerr/CFT Correspondence, JHEP 10 (2012) 182 [arXiv:1208.3322] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    U. Khanal, Rotating black hole in asymptotic de Sitter space: perturbation of the space-time with spin fields, Phys. Rev. D 28 (1983) 1291 [INSPIRE].
  9. [9]
    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].
  10. [10]
    D. Batic and H. Schmid, Heun equation, Teukolsky equation and type-D metrics, J. Math. Phys. 48 (2007) 042502 [gr-qc/0701064] [INSPIRE].
  11. [11]
    H. Suzuki, E. Takasugi and H. Umetsu, Perturbations of Kerr-de Sitter black hole and Heuns equations, Prog. Theor. Phys. 100 (1998) 491 [gr-qc/9805064] [INSPIRE].
  12. [12]
    A. Ronveaux and F. Arscott, Heuns differential equations, Oxford University Press (1995).Google Scholar
  13. [13]
    R. Garnier, Sur des équations différentielles du troisième ordre dont lintégrale générale est uniforme et sur une classe déquations nouvelles dordre supérieur dont lintégrale générale à ses points critiques fixes, Annales Sci. Ecole Norm. Sup. 29 (1912) 1.Google Scholar
  14. [14]
    K. Iwasaki, H. Kimura, S. Shimomura and M. Yoshida, From Gauss to Painlevé: A Modern Theory of Special Functions, in Aspects of Mathematics E, vol. 16, Braunschweig (1991).Google Scholar
  15. [15]
    A. Litvinov, S. Lukyanov, N. Nekrasov and A. Zamolodchikov, Classical Conformal Blocks and Painleve VI, JHEP 07 (2014) 144 [arXiv:1309.4700] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    M. Lencsés and F. Novaes, Classical Conformal Blocks and Accessory Parameters from Isomonodromic Deformations, JHEP 04 (2018) 096 [arXiv:1709.03476] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    T. Anselmo, R. Nelson, B.C. da Cunha and D.G. Crowdy, Accessory parameters in conformal mapping: exploiting the isomonodromic tau function for Painlevé VI, Proc. Roy. Soc. Lond. A 474 (2018) 20180080.Google Scholar
  18. [18]
    P. Menotti, On the monodromy problem for the four-punctured sphere, J. Phys. A 47 (2014) 415201 [arXiv:1401.2409] [INSPIRE].
  19. [19]
    L. Hollands and O. Kidwai, Higher length-twist coordinates, generalized Heuns opers and twisted superpotentials, arXiv:1710.04438 [INSPIRE].
  20. [20]
    D. Anninos and T. Anous, A de Sitter Hoedown, JHEP 08 (2010) 131 [arXiv:1002.1717] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    S. Yoshida, N. Uchikata and T. Futamase, Quasinormal modes of Kerr-de Sitter black holes, Phys. Rev. D 81 (2010) 044005 [INSPIRE].
  22. [22]
    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].
  23. [23]
    E.W. Leaver, An Analytic representation for the quasi normal modes of Kerr black holes, Proc. Roy. Soc. Lond. A 402 (1985) 285 [INSPIRE].
  24. [24]
    E.W. Leaver, Solutions to a generalized spheroidal wave equation: Teukolskys equations in general relativity, and the two-center problem in molecular quantum mechanics, J. Math. Phys. 27 (1986) 1238.Google Scholar
  25. [25]
    O.J. Tattersall, Kerr-(anti-)de Sitter black holes: Perturbations and quasinormal modes in the slow rotation limit, Phys. Rev. D 98 (2018) 104013 [arXiv:1808.10758] [INSPIRE].
  26. [26]
    S. Dyatlov, Asymptotic distribution of quasi-normal modes for Kerr-de Sitter black holes, Annales Henri Poincaré 13 (2012) 1101 [arXiv:1101.1260] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    O.J.C. Dias, F.C. Eperon, H.S. Reall and J.E. Santos, Strong cosmic censorship in de Sitter space, Phys. Rev. D 97 (2018) 104060 [arXiv:1801.09694] [INSPIRE].
  28. [28]
    F. Novaes and B. Carneiro da Cunha, Isomonodromy, Painlevé transcendents and scattering off of black holes, JHEP 07 (2014) 132 [arXiv:1404.5188] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  29. [29]
    B. Carter, Black holes (les astres occlus), in Black Hole Equilibrium States, Gordon and Breach Science Publishers, NY (1973).Google Scholar
  30. [30]
    F. Mellor and I. Moss, Stability of Black Holes in de Sitter Space, Phys. Rev. D 41 (1990) 403 [INSPIRE].
  31. [31]
    S. Akcay and R.A. Matzner, Kerr-de Sitter Universe, Class. Quant. Grav. 28 (2011) 085012 [arXiv:1011.0479] [INSPIRE].
  32. [32]
    I.S. Booth and R.B. Mann, Cosmological pair production of charged and rotating black holes, Nucl. Phys. B 539 (1999) 267 [gr-qc/9806056] [INSPIRE].
  33. [33]
    P. Hintz and A. Vasy, The global non-linear stability of the Kerr-de Sitter family of black holes, arXiv:1606.04014 [INSPIRE].
  34. [34]
    A. Castro, J.M. Lapan, A. Maloney and M.J. Rodriguez, Black Hole Scattering from Monodromy, Class. Quant. Grav. 30 (2013) 165005 [arXiv:1304.3781] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    B. Carneiro da Cunha and F. Novaes, Kerr-de Sitter greybody factors via isomonodromy, Phys. Rev. D 93 (2016) 024045 [arXiv:1508.04046] [INSPIRE].
  36. [36]
    S.Y. Slavyanov and W. Lay, Special Functions: A Unified Theory Based on Singularities, Oxford University Press, U.S.A. (2000).Google Scholar
  37. [37]
    B. Carneiro da Cunha and F. Novaes, Kerr Scattering Coefficients via Isomonodromy, JHEP 11 (2015) 144 [arXiv:1506.06588] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    A.M. Ghezelbash and R.B. Mann, Entropy and mass bounds of Kerr-de Sitter spacetimes, Phys. Rev. D 72 (2005) 064024 [hep-th/0412300] [INSPIRE].
  39. [39]
    S.L. Detweiler, Black holes and gravitational waves. III. The resonant frequencies of rotating holes, Astrophys. J. 239 (1980) 292 [INSPIRE].
  40. [40]
    T. Harmark, J. Natario and R. Schiappa, Greybody Factors for d-Dimensional Black Holes, Adv. Theor. Math. Phys. 14 (2010) 727 [arXiv:0708.0017] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  41. [41]
    H. Yang, A. Zimmerman, A. Zenginoğlu, F. Zhang, E. Berti and Y. Chen, Quasinormal modes of nearly extremal Kerr spacetimes: spectrum bifurcation and power-law ringdown, Phys. Rev. D 88 (2013) 044047 [arXiv:1307.8086] [INSPIRE].
  42. [42]
    A. Castro, A. Maloney and A. Strominger, Hidden Conformal Symmetry of the Kerr Black Hole, Phys. Rev. D 82 (2010) 024008 [arXiv:1004.0996] [INSPIRE].
  43. [43]
    L.C.B. Crispino, A. Higuchi, E.S. Oliveira and J.V. Rocha, Greybody factors for nonminimally coupled scalar fields in Schwarzschild-de Sitter spacetime, Phys. Rev. D 87 (2013) 104034 [arXiv:1304.0467] [INSPIRE].
  44. [44]
    S. Mano, H. Suzuki and E. Takasugi, Analytic solutions of the Teukolsky equation and their low frequency expansions, Prog. Theor. Phys. 95 (1996) 1079 [gr-qc/9603020] [INSPIRE].
  45. [45]
    M. Sasaki and H. Tagoshi, Analytic black hole perturbation approach to gravitational radiation, Living Rev. Rel. 6 (2003) 6 [gr-qc/0306120] [INSPIRE].
  46. [46]
    M. Casals and P. Zimmerman, Perturbations of Extremal Kerr Spacetime: Analytic Framework and Late-time Tails, arXiv:1801.05830 [INSPIRE].
  47. [47]
    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].
  48. [48]
    H. Suzuki, E. Takasugi and H. Umetsu, Absorption rate of the Kerr-de Sitter black hole and the Kerr-Newman-de Sitter black hole, Prog. Theor. Phys. 103 (2000) 723 [gr-qc/9911079] [INSPIRE].
  49. [49]
    L. Motl and A. Neitzke, Asymptotic black hole quasinormal frequencies, Adv. Theor. Math. Phys. 7 (2003) 307 [hep-th/0301173] [INSPIRE].MathSciNetCrossRefGoogle Scholar
  50. [50]
    M. Casals and A.C. Ottewill, Spin-1 quasinormal frequencies in Schwarzschild spacetime for large overtone number, Phys. Rev. D 97 (2018) 024048 [arXiv:1606.03423] [INSPIRE].
  51. [51]
    T. Tachizawa and K.-i. Maeda, Superradiance in the kerr-de sitter space-time, Phys. Lett. A 172 (1993) 325.Google Scholar
  52. [52]
    M. Casals, C. Kavanagh and A.C. Ottewill, High-order late-time tail in a Kerr spacetime, Phys. Rev. D 94 (2016) 124053 [arXiv:1608.05392] [INSPIRE].
  53. [53]
    B.E. Barrowes, K. O’Neill, G.T.M. and J.A. Kong, On the Asymptotic Expansion of the Spheroidal Wave Function and Its Eigenvalues for Complex Size Parameter, Stud. Appl. Math. 113 (2004) 271.Google Scholar
  54. [54]
    E.S.C. Ching, P.T. Leung, W.M. Suen and K. Young, Wave propagation in gravitational systems: Late time behavior, Phys. Rev. D 52 (1995) 2118 [gr-qc/9507035] [INSPIRE].
  55. [55]
    K. Glampedakis and N. Andersson, Late time dynamics of rapidly rotating black holes, Phys. Rev. D 64 (2001) 104021 [gr-qc/0103054] [INSPIRE].
  56. [56]
    L. Barack and A. Ori, Late time decay of scalar perturbations outside rotating black holes, Phys. Rev. Lett. 82 (1999) 4388 [gr-qc/9902082] [INSPIRE].
  57. [57]
    S. Hod, The Radiative tail of realistic gravitational collapse, Phys. Rev. Lett. 84 (2000) 10 [gr-qc/9907096] [INSPIRE].
  58. [58]
    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].
  59. [59]
    S.E. Gralla and P. Zimmerman, Critical Exponents of Extremal Kerr Perturbations, Class. Quant. Grav. 35 (2018) 095002 [arXiv:1711.00855] [INSPIRE].
  60. [60]
    S. Aretakis, Horizon Instability of Extremal Black Holes, Adv. Theor. Math. Phys. 19 (2015) 507 [arXiv:1206.6598] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  61. [61]
    S. Hod, Slow relaxation of rapidly rotating black holes, Phys. Rev. D 78 (2008) 084035 [arXiv:0811.3806] [INSPIRE].
  62. [62]
    D. Anninos and T. Hartman, Holography at an Extremal de Sitter Horizon, JHEP 03 (2010) 096 [arXiv:0910.4587] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  63. [63]
    NIST Digital Library of Mathematical Functions, Release 1.0.5, 2012-10-01 [].
  64. [64]
    V. Cardoso and J.P.S. Lemos, Quasinormal modes of the near extremal Schwarzschild-de Sitter black hole, Phys. Rev. D 67 (2003) 084020 [gr-qc/0301078] [INSPIRE].
  65. [65]
    M. Jimbo, T. Miwa and A.K. Ueno, Monodromy Preserving Deformation of Linear Ordinary Differential Equations With Rational Coefficients. I, Physica D 2 (1981) 306.Google Scholar
  66. [66]
    M. Jimbo and T. Miwa, Monodromy Preserving Deformation of Linear Ordinary Differential Equations with Rational Coefficients. II, Physica D 2 (1981) 407.Google Scholar
  67. [67]
    M. Jimbo and T. Miwa, Monodromy Preserving Deformation of Linear Ordinary Differential Equations with Rational Coefficients. III, Physica D 4 (1981) 26.Google Scholar
  68. [68]
    M. Casals and C. Marinho, in preparation.Google Scholar
  69. [69]
    P. Menotti, Classical conformal blocks, Mod. Phys. Lett. A 31 (2016) 1650159 [arXiv:1601.04457] [INSPIRE].
  70. [70]
    N. Iorgov, O. Lisovyy and Yu. Tykhyy, Painlevé VI connection problem and monodromy of c = 1 conformal blocks, JHEP 12 (2013) 029 [arXiv:1308.4092] [INSPIRE].
  71. [71]
    J.B. Amado, B. Carneiro da Cunha and E. Pallante, On the Kerr-AdS/CFT correspondence, JHEP 08 (2017) 094 [arXiv:1702.01016] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  72. [72]
    J. Barragán-Amado, B. Carneiro Da Cunha and E. Pallante, Scalar quasinormal modes of Kerr-AdS 5, arXiv:1812.08921 [INSPIRE].
  73. [73]
    O. Gamayun, N. Iorgov and O. Lisovyy, How instanton combinatorics solves Painlevé VI, V and IIIs, J. Phys. A 46 (2013) 335203 [arXiv:1302.1832] [INSPIRE].
  74. [74]
    O. Lisovyy, H. Nagoya and J. Roussillon, Irregular conformal blocks and connection formulae for Painlevé V functions, J. Math. Phys. 59 (2018) 091409 [arXiv:1806.08344] [INSPIRE].
  75. [75]
    N. Iorgov, O. Lisovyy and J. Teschner, Isomonodromic tau-functions from Liouville conformal blocks, Commun. Math. Phys. 336 (2015) 671 [arXiv:1401.6104] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  76. [76]
    P. Gavrylenko and O. Lisovyy, Fredholm Determinant and Nekrasov Sum Representations of Isomonodromic Tau Functions, Commun. Math. Phys. 363 (2018) 1 [arXiv:1608.00958] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  77. [77]
    V. Frolov, P. Krtous and D. Kubiznak, Black holes, hidden symmetries and complete integrability, Living Rev. Rel. 20 (2017) 6 [arXiv:1705.05482] [INSPIRE].CrossRefGoogle Scholar
  78. [78]
    M. Guica, T. Hartman, W. Song and A. Strominger, The Kerr/CFT Correspondence, Phys. Rev. D 80 (2009) 124008 [arXiv:0809.4266] [INSPIRE].
  79. [79]
    G. Compère, The Kerr/CFT correspondence and its extensions, Living Rev. Rel. 15 (2012) 11 [arXiv:1203.3561] [INSPIRE].CrossRefzbMATHGoogle Scholar
  80. [80]
    B. Carneiro da Cunha and M. Guica, Exploring the BTZ bulk with boundary conformal blocks, arXiv:1604.07383 [INSPIRE].
  81. [81]
    M. Piatek and A.R. Pietrykowski, Solving Heuns equation using conformal blocks, Nucl. Phys. B 938 (2019) 543 [arXiv:1708.06135] [INSPIRE].
  82. [82]
    D. Braak, Integrability of the rabi model, Phys. Rev. Lett. 107 (2011) 100401 [arXiv:1103.2461].ADSCrossRefGoogle Scholar
  83. [83]
    B. Carneiro da Cunha, M.C. de Almeida and A.R. de Queiroz, On the Existence of Monodromies for the Rabi model, arXiv:1508.01342 [INSPIRE].
  84. [84]
    O. Gamayun, N. Iorgov and O. Lisovyy, Conformal field theory of Painlevé VI, JHEP 10 (2012) 038 [Erratum ibid. 10 (2012) 183] [arXiv:1207.0787] [INSPIRE].
  85. [85]
    L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2019

Authors and Affiliations

  • Fábio Novaes
    • 1
    Email author
  • Cássio I. S. Marinho
    • 2
  • Máté Lencsés
    • 1
  • Marc Casals
    • 2
    • 3
  1. 1.International Institute of PhysicsFederal University of Rio Grande do Norte, Campus UniversitárioNatalBrazil
  2. 2.Centro Brasileiro de Pesquisas Físicas (CBPF)Rio de JaneiroBrazil
  3. 3.School of Mathematics and StatisticsUniversity College DublinDublin 4Ireland

Personalised recommendations