Advertisement

Stationary resonances of rapidly-rotating Kerr black holes

Regular Article - Theoretical Physics

Abstract

The Klein–Gordon equation for a massive scalar field in the background of a rapidly-rotating Kerr black hole is studied analytically. In particular, we derive a simple formula for the stationary (marginally-stable) resonances of the field in the black-hole spacetime. The analytically derived formula is shown to agree with direct numerical computations of the resonances. Our results provide an upper bound on the instability regime of rapidly rotating Kerr black holes to massive scalar perturbations.

Keywords

Black Hole Stationary Resonance Gordon Equation Kerr Black Hole Radial Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

This research is supported by the Carmel Science Foundation. I thank Yael Oren, Arbel M. Ongo and Ayelet B. Lata for stimulating discussions.

References

  1. 1.
    R. Ruffini, J.A. Wheeler, Phys. Today 24, 30 (1971) ADSCrossRefGoogle Scholar
  2. 2.
    B. Carter, Black holes, in Proceedings of 1972 Session of Ecole d’ete de Physique Theorique, ed. by C. De Witt, B.S. De Witt (Gordon and Breach, New York, 1973) Google Scholar
  3. 3.
    J.D. Bekenstein, Phys. Today 33, 24 (1980) ADSCrossRefGoogle Scholar
  4. 4.
    D. Núñez, H. Quevedo, D. Sudarsky, Phys. Rev. Lett. 76, 571 (1996) MathSciNetADSMATHCrossRefGoogle Scholar
  5. 5.
    S. Hod, Phys. Rev. D 84, 124030 (2011). arXiv:1112.3286 ADSCrossRefGoogle Scholar
  6. 6.
    J.E. Chase, Commun. Math. Phys. 19, 276 (1970) MathSciNetADSMATHCrossRefGoogle Scholar
  7. 7.
    J.D. Bekenstein, Phys. Rev. Lett. 28, 452 (1972) ADSCrossRefGoogle Scholar
  8. 8.
    C. Teitelboim, Lett. Nuovo Cimento 3, 326 (1972) CrossRefGoogle Scholar
  9. 9.
    I. Pena, D. Sudarsky, Class. Quantum Gravity 14, 3131 (1997) MathSciNetADSMATHCrossRefGoogle Scholar
  10. 10.
    J.D. Bekenstein, Phys. Rev. D 5, 1239 (1972) MathSciNetADSCrossRefGoogle Scholar
  11. 11.
    J.D. Bekenstein, Phys. Rev. D 5, 2403 (1972) MathSciNetADSCrossRefGoogle Scholar
  12. 12.
    M. Heusler, J. Math. Phys. 33, 3497 (1992) MathSciNetADSMATHCrossRefGoogle Scholar
  13. 13.
    D. Sudarsky, Class. Quantum Gravity 12, 579 (1995) MathSciNetADSMATHCrossRefGoogle Scholar
  14. 14.
    J. Hartle, Phys. Rev. D 3, 2938 (1971) ADSCrossRefGoogle Scholar
  15. 15.
    C. Teitelboim, Lett. Nuovo Cimento 3, 397 (1972) ADSCrossRefGoogle Scholar
  16. 16.
    H.P. Nollert, Class. Quantum Gravity 16, R159 (1999) MathSciNetADSMATHCrossRefGoogle Scholar
  17. 17.
    E. Berti, V. Cardoso, A.O. Starinets, Class. Quantum Gravity 26, 163001 (2009) MathSciNetADSCrossRefGoogle Scholar
  18. 18.
    E.W. Leaver, Proc. R. Soc. A 402, 285 (1985) MathSciNetADSCrossRefGoogle Scholar
  19. 19.
    B. Mashhoon, Phys. Rev. D 31, 290 (1985) MathSciNetADSCrossRefGoogle Scholar
  20. 20.
    H.P. Nollert, Phys. Rev. D 47, 5253 (1993) ADSCrossRefGoogle Scholar
  21. 21.
    S. Hod, Phys. Rev. Lett. 81, 4293 (1998). arXiv:gr-qc/9812002 MathSciNetADSCrossRefGoogle Scholar
  22. 22.
    G.T. Horowitz, V.E. Hubeny, Phys. Rev. D 62, 024027 (2000) MathSciNetADSCrossRefGoogle Scholar
  23. 23.
    K. Glampedakis, N. Andersson, Phys. Rev. D 64, 104021 (2001) MathSciNetADSCrossRefGoogle Scholar
  24. 24.
    U. Keshet, S. Hod, Phys. Rev. D 76, R061501 (2007). arXiv:0705.1179 MathSciNetADSCrossRefGoogle Scholar
  25. 25.
    S. Hod, Phys. Rev. D 75, 064013 (2007). arXiv:gr-qc/0611004 MathSciNetADSCrossRefGoogle Scholar
  26. 26.
    S. Hod, Class. Quantum Gravity 24, 4235 (2007). arXiv:0705.2306 MathSciNetADSMATHCrossRefGoogle Scholar
  27. 27.
    A. Gruzinov. arXiv:0705.1725 [gr-qc]
  28. 28.
    A. Pesci, Class. Quantum Gravity 24, 6219 (2007) MathSciNetADSMATHCrossRefGoogle Scholar
  29. 29.
    S. Hod, Phys. Rev. D 78, 084035 (2008). arXiv:0811.3806 MathSciNetADSCrossRefGoogle Scholar
  30. 30.
    S. Hod, Phys. Lett. B 666, 483 (2008). arXiv:0810.5419 MathSciNetADSCrossRefGoogle Scholar
  31. 31.
    S. Hod, Phys. Rev. D 80, 064004 (2009). arXiv:0909.0314 ADSCrossRefGoogle Scholar
  32. 32.
    V. Cardoso, A.S. Miranda, E. Berti, H. Witek, V.T. Zanchin, Phys. Rev. D 79, 064016 (2009) MathSciNetADSCrossRefGoogle Scholar
  33. 33.
    S. Hod, Phys. Lett. A 374, 2901 (2010). arXiv:1006.4439 ADSMATHCrossRefGoogle Scholar
  34. 34.
    R.A. Konoplya, A. Zhidenko, Rev. Mod. Phys. 83, 793 (2011) ADSCrossRefGoogle Scholar
  35. 35.
    S. Hod, Phys. Rev. D 84, 044046 (2011). arXiv:1109.4080 ADSCrossRefGoogle Scholar
  36. 36.
    Y. Décanini, A. Folacci, B. Raffaelli, Phys. Rev. D 84, 084035 (2011) ADSCrossRefGoogle Scholar
  37. 37.
    S. Hod, Phys. Lett. B 710, 349 (2012). arXiv:1205.5087 MathSciNetADSCrossRefGoogle Scholar
  38. 38.
    S. Hod, Phys. Lett. B 715, 348 (2012). arXiv:1207.5282 ADSCrossRefGoogle Scholar
  39. 39.
    R.H. Price, Phys. Rev. D 5, 2419 (1972) MathSciNetADSCrossRefGoogle Scholar
  40. 40.
    C. Gundlach, R.H. Price, J. Pullin, Phys. Rev. D 49, 883 (1994) ADSCrossRefGoogle Scholar
  41. 41.
    J. Bicák, Gen. Relativ. Gravit. 3, 331 (1972) ADSCrossRefGoogle Scholar
  42. 42.
    E.S.C. Ching, P.T. Leung, W.M. Suen, K. Young, Phys. Rev. Lett. 74, 2414 (1995) ADSCrossRefGoogle Scholar
  43. 43.
    E.S.C. Ching, P.T. Leung, W.M. Suen, K. Young, Phys. Rev. D 52, 2118 (1995) ADSCrossRefGoogle Scholar
  44. 44.
    S. Hod, T. Piran, Phys. Rev. D 58, 024017 (1998). arXiv:gr-qc/9712041 MathSciNetADSCrossRefGoogle Scholar
  45. 45.
    S. Hod, T. Piran, Phys. Rev. D 58, 024018 (1998). arXiv:gr-qc/9801001 MathSciNetADSCrossRefGoogle Scholar
  46. 46.
    S. Hod, T. Piran, Phys. Rev. D 58, 044018 (1998). arXiv:gr-qc/9801059 ADSCrossRefGoogle Scholar
  47. 47.
    S. Hod, T. Piran, Phys. Rev. D 58, 024019 (1998). arXiv:gr-qc/9801060 MathSciNetADSCrossRefGoogle Scholar
  48. 48.
    S. Hod, Phys. Rev. D 58, 104022 (1998). arXiv:gr-qc/9811032 MathSciNetADSCrossRefGoogle Scholar
  49. 49.
    S. Hod, Phys. Rev. D 61, 024033 (2000). arXiv:gr-qc/9902072 MathSciNetADSCrossRefGoogle Scholar
  50. 50.
    S. Hod, Phys. Rev. D 61, 064018 (2000). arXiv:gr-qc/9902073 MathSciNetADSCrossRefGoogle Scholar
  51. 51.
    L. Barack, Phys. Rev. D 61, 024026 (2000) MathSciNetADSCrossRefGoogle Scholar
  52. 52.
    S. Hod, Phys. Rev. Lett. 84, 10 (2000). arXiv:gr-qc/9907096 ADSCrossRefGoogle Scholar
  53. 53.
    S. Hod, Phys. Rev. D 60, 104053 (1999). arXiv:gr-qc/9907044 MathSciNetADSCrossRefGoogle Scholar
  54. 54.
    S. Hod, Class. Quantum Gravity 26, 028001 (2009). arXiv:0902.0237 MathSciNetADSCrossRefGoogle Scholar
  55. 55.
    S. Hod, Class. Quantum Gravity 18, 1311 (2001). arXiv:gr-qc/0008001 MathSciNetADSMATHCrossRefGoogle Scholar
  56. 56.
    S. Hod, Phys. Rev. D 66, 024001 (2002). arXiv:gr-qc/0201017 MathSciNetADSCrossRefGoogle Scholar
  57. 57.
    R.J. Gleiser, R.H. Price, J. Pullin, Class. Quantum Gravity 25, 072001 (2008) MathSciNetADSCrossRefGoogle Scholar
  58. 58.
    M. Tiglio, L.E. Kidder, S.A. Teukolsky, Class. Quantum Gravity 25, 105022 (2008) ADSCrossRefGoogle Scholar
  59. 59.
    R. Moderski, M. Rogatko, Phys. Rev. D 77, 124007 (2008) ADSCrossRefGoogle Scholar
  60. 60.
    X. He, J. Jing, Nucl. Phys. B 755, 313 (2006) MathSciNetADSMATHCrossRefGoogle Scholar
  61. 61.
    H. Koyama, A. Tomimatsu, Phys. Rev. D 65, 084031 (2002) MathSciNetADSCrossRefGoogle Scholar
  62. 62.
    B. Wang, C. Molina, E. Abdalla, Phys. Rev. D 63, 084001 (2001) ADSCrossRefGoogle Scholar
  63. 63.
    A. Zenginoglu, M. Tiglio, Phys. Rev. D 80, 024044 (2009) MathSciNetADSCrossRefGoogle Scholar
  64. 64.
    A.J. Amsel, G.T. Horowitz, D. Marolf, M.M. Roberts, J. High Energy Phys. 0909, 044 (2009) MathSciNetADSCrossRefGoogle Scholar
  65. 65.
    Ya.B. Zel’dovich, Pis’ma Zh. Eksp. Teor. Fiz. 14, 270 (1971). JETP Lett. 14, 180 (1971) Google Scholar
  66. 66.
    Ya.B. Zel’dovich, Zh. Eksp. Teor. Fiz. 62, 2076 (1972). Sov. Phys. JETP, 35, 1085 (1972) Google Scholar
  67. 67.
    W.H. Press, S.A. Teukolsky, Astrophys. J. 185, 649 (1973) MathSciNetADSCrossRefGoogle Scholar
  68. 68.
    W.H. Press, S.A. Teukolsky, Nature 238, 211 (1972) ADSCrossRefGoogle Scholar
  69. 69.
    T. Damour, N. Deruelle, R. Ruffini, Lett. Nuovo Cimento 15, 257 (1976) ADSCrossRefGoogle Scholar
  70. 70.
    T.M. Zouros, D.M. Eardley, Ann. Phys. 118, 139 (1979) ADSCrossRefGoogle Scholar
  71. 71.
    S. Detweiler, Phys. Rev. D 22, 2323 (1980) ADSCrossRefGoogle Scholar
  72. 72.
    H. Furuhashi, Y. Nambu, Prog. Theor. Phys. 112, 983 (2004) ADSMATHCrossRefGoogle Scholar
  73. 73.
    V. Cardoso, O.J.C. Dias, J.P.S. Lemos, S. Yoshida, Phys. Rev. D 70, 044039 (2004) MathSciNetADSCrossRefGoogle Scholar
  74. 74.
    V. Cardoso, O.J.C. Dias, J.P.S. Lemos, S. Yoshida, Phys. Rev. D 70, 049903 (2004). Erratum MathSciNetADSCrossRefGoogle Scholar
  75. 75.
    V. Cardoso, J.P.S. Lemos, Phys. Lett. B 621, 219 (2005) MathSciNetADSMATHCrossRefGoogle Scholar
  76. 76.
    V. Cardoso, S. Yoshida, J. High Energy Phys. 0507, 009 (2005) MathSciNetADSCrossRefGoogle Scholar
  77. 77.
    H. Witek, V. Cardoso, A. Ishibashi, U. Sperhake. arXiv:1212.0551
  78. 78.
    S.R. Dolan, Phys. Rev. D 76, 084001 (2007) MathSciNetADSCrossRefGoogle Scholar
  79. 79.
    S. Hod, O. Hod, Phys. Rev. D 81, 061502 (2010). Rapid communication. arXiv:0910.0734 MathSciNetADSCrossRefGoogle Scholar
  80. 80.
    S. Hod, O. Hod. arXiv:0912.2761
  81. 81.
    H.R. Beyer, J. Math. Phys. 52, 102502 (2011) MathSciNetADSCrossRefGoogle Scholar
  82. 82.
    S. Hod, Phys. Lett. B 708, 320 (2012). arXiv:1205.1872 ADSCrossRefGoogle Scholar
  83. 83.
    S. Hod, Phys. Lett. B 713, 505 (2012) MathSciNetADSCrossRefGoogle Scholar
  84. 84.
    S. Hod, Phys. Rev. D 86, 104026 (2012). arXiv:1211.3202 ADSCrossRefGoogle Scholar
  85. 85.
    S. Hod, Phys. Lett. B 718, 1489 (2012) MathSciNetADSGoogle Scholar
  86. 86.
    S.R. Dolan. arXiv:1212.1477
  87. 87.
    S. Chandrasekhar, The Mathematical Theory of Black Holes (Oxford University Press, New York, 1983) MATHGoogle Scholar
  88. 88.
    R.P. Kerr, Phys. Rev. Lett. 11, 237 (1963) MathSciNetADSMATHCrossRefGoogle Scholar
  89. 89.
    S.A. Teukolsky, Phys. Rev. Lett. 29, 1114 (1972) ADSCrossRefGoogle Scholar
  90. 90.
    S.A. Teukolsky, Astrophys. J. 185, 635 (1973) MathSciNetADSCrossRefGoogle Scholar
  91. 91.
    A. Ronveaux, Heun’s Differential Equations (Oxford University Press, Oxford, 1995) MATHGoogle Scholar
  92. 92.
    C. Flammer, Spheroidal Wave Functions (Stanford University Press, Stanford, 1957) MATHGoogle Scholar
  93. 93.
    P.P. Fiziev. arXiv:0902.1277
  94. 94.
    R.S. Borissov, P.P. Fiziev. arXiv:0903.3617
  95. 95.
    P.P. Fiziev, Phys. Rev. D 80, 124001 (2009) MathSciNetADSCrossRefGoogle Scholar
  96. 96.
    P.P. Fiziev, Class. Quantum Gravity 27, 135001 (2010) MathSciNetADSCrossRefGoogle Scholar
  97. 97.
    M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1970) Google Scholar
  98. 98.
    T. Hartman, W. Song, A. Strominger, J. High Energy Phys. 1003, 118 (2010) MathSciNetADSCrossRefGoogle Scholar
  99. 99.
    S. Hod, Phys. Rev. Lett. 100, 121101 (2008). arXiv:0805.3873 MathSciNetADSCrossRefGoogle Scholar
  100. 100.
    S. Hod, Phys. Lett. B 717, 462 (2012) ADSCrossRefGoogle Scholar
  101. 101.
    L.E. Simone, C.M. Will, Class. Quantum Gravity 9, 963 (1992) MathSciNetADSCrossRefGoogle Scholar
  102. 102.
    P.M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953) MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Società Italiana di Fisica 2013

Authors and Affiliations

  1. 1.The Ruppin Academic CenterEmeq HeferIsrael
  2. 2.The Hadassah InstituteJerusalemIsrael

Personalised recommendations