Skip to main content
SpringerLink
Log in

Stationary resonances of rapidly-rotating Kerr black holes

  • Regular Article - Theoretical Physics
  • Published:
The European Physical Journal C Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. This statement seems to hold true for all bosonic fields.

  2. In these units μ has the dimensions of 1/length.

  3. We note that, in addition to the bound states of the massive field, the field also has an infinite set of discrete quasinormal resonances [101] which are characterized by outgoing waves at spatial infinity.

  4. In the τ≪1 regime the condition 2β>1 is satisfied by modes with \(m<\sqrt{2l(l+1)/3}\), see Eq. (28) below. In particular, it is satisfied by the fundamental l=m=1 mode.

  5. We denote by ϵ the solution of the quadratic equation (30). The O(ϵ 3) term in (30) is −16(2n+1)ϵ 3. In the τ→0 limit this term introduces a small correction of the form: ϵ→ϵ(1+δ) with \(\delta\simeq{{m^{2}(2n+1)}\over{\ell(\ell+1+2n)^{2}}}\). Note that in the regime 2β>1 that we consider (\(m<\sqrt{2l(l+1)/3}\), see footnote 4), one has \(\delta<{{{2\over 3}l(l+1)}\over{\sqrt{{4\over 3}l(l+1)+1}[\sqrt{{4\over 3}l(l+1)+1}+1]^{2}}}\ll 1\).

  6. Note that \({\bar{\epsilon}}^{2}\ll 1\) in the regime 2β>1 that we consider (\(m<\sqrt{2l(l+1)/3}\), see footnote 4). This justifies the perturbation expansion (11).

References

  1. R. Ruffini, J.A. Wheeler, Phys. Today 24, 30 (1971)

    Article  ADS  Google Scholar 

  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. J.D. Bekenstein, Phys. Today 33, 24 (1980)

    Article  ADS  Google Scholar 

  4. D. Núñez, H. Quevedo, D. Sudarsky, Phys. Rev. Lett. 76, 571 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  5. S. Hod, Phys. Rev. D 84, 124030 (2011). arXiv:1112.3286

    Article  ADS  Google Scholar 

  6. J.E. Chase, Commun. Math. Phys. 19, 276 (1970)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. J.D. Bekenstein, Phys. Rev. Lett. 28, 452 (1972)

    Article  ADS  Google Scholar 

  8. C. Teitelboim, Lett. Nuovo Cimento 3, 326 (1972)

    Article  Google Scholar 

  9. I. Pena, D. Sudarsky, Class. Quantum Gravity 14, 3131 (1997)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  10. J.D. Bekenstein, Phys. Rev. D 5, 1239 (1972)

    Article  MathSciNet  ADS  Google Scholar 

  11. J.D. Bekenstein, Phys. Rev. D 5, 2403 (1972)

    Article  MathSciNet  ADS  Google Scholar 

  12. M. Heusler, J. Math. Phys. 33, 3497 (1992)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. D. Sudarsky, Class. Quantum Gravity 12, 579 (1995)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  14. J. Hartle, Phys. Rev. D 3, 2938 (1971)

    Article  ADS  Google Scholar 

  15. C. Teitelboim, Lett. Nuovo Cimento 3, 397 (1972)

    Article  ADS  Google Scholar 

  16. H.P. Nollert, Class. Quantum Gravity 16, R159 (1999)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  17. E. Berti, V. Cardoso, A.O. Starinets, Class. Quantum Gravity 26, 163001 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  18. E.W. Leaver, Proc. R. Soc. A 402, 285 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  19. B. Mashhoon, Phys. Rev. D 31, 290 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  20. H.P. Nollert, Phys. Rev. D 47, 5253 (1993)

    Article  ADS  Google Scholar 

  21. S. Hod, Phys. Rev. Lett. 81, 4293 (1998). arXiv:gr-qc/9812002

    Article  MathSciNet  ADS  Google Scholar 

  22. G.T. Horowitz, V.E. Hubeny, Phys. Rev. D 62, 024027 (2000)

    Article  MathSciNet  ADS  Google Scholar 

  23. K. Glampedakis, N. Andersson, Phys. Rev. D 64, 104021 (2001)

    Article  MathSciNet  ADS  Google Scholar 

  24. U. Keshet, S. Hod, Phys. Rev. D 76, R061501 (2007). arXiv:0705.1179

    Article  MathSciNet  ADS  Google Scholar 

  25. S. Hod, Phys. Rev. D 75, 064013 (2007). arXiv:gr-qc/0611004

    Article  MathSciNet  ADS  Google Scholar 

  26. S. Hod, Class. Quantum Gravity 24, 4235 (2007). arXiv:0705.2306

    Article  MathSciNet  ADS  MATH  Google Scholar 

  27. A. Gruzinov. arXiv:0705.1725 [gr-qc]

  28. A. Pesci, Class. Quantum Gravity 24, 6219 (2007)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  29. S. Hod, Phys. Rev. D 78, 084035 (2008). arXiv:0811.3806

    Article  MathSciNet  ADS  Google Scholar 

  30. S. Hod, Phys. Lett. B 666, 483 (2008). arXiv:0810.5419

    Article  MathSciNet  ADS  Google Scholar 

  31. S. Hod, Phys. Rev. D 80, 064004 (2009). arXiv:0909.0314

    Article  ADS  Google Scholar 

  32. V. Cardoso, A.S. Miranda, E. Berti, H. Witek, V.T. Zanchin, Phys. Rev. D 79, 064016 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  33. S. Hod, Phys. Lett. A 374, 2901 (2010). arXiv:1006.4439

    Article  ADS  MATH  Google Scholar 

  34. R.A. Konoplya, A. Zhidenko, Rev. Mod. Phys. 83, 793 (2011)

    Article  ADS  Google Scholar 

  35. S. Hod, Phys. Rev. D 84, 044046 (2011). arXiv:1109.4080

    Article  ADS  Google Scholar 

  36. Y. Décanini, A. Folacci, B. Raffaelli, Phys. Rev. D 84, 084035 (2011)

    Article  ADS  Google Scholar 

  37. S. Hod, Phys. Lett. B 710, 349 (2012). arXiv:1205.5087

    Article  MathSciNet  ADS  Google Scholar 

  38. S. Hod, Phys. Lett. B 715, 348 (2012). arXiv:1207.5282

    Article  ADS  Google Scholar 

  39. R.H. Price, Phys. Rev. D 5, 2419 (1972)

    Article  MathSciNet  ADS  Google Scholar 

  40. C. Gundlach, R.H. Price, J. Pullin, Phys. Rev. D 49, 883 (1994)

    Article  ADS  Google Scholar 

  41. J. Bicák, Gen. Relativ. Gravit. 3, 331 (1972)

    Article  ADS  Google Scholar 

  42. E.S.C. Ching, P.T. Leung, W.M. Suen, K. Young, Phys. Rev. Lett. 74, 2414 (1995)

    Article  ADS  Google Scholar 

  43. E.S.C. Ching, P.T. Leung, W.M. Suen, K. Young, Phys. Rev. D 52, 2118 (1995)

    Article  ADS  Google Scholar 

  44. S. Hod, T. Piran, Phys. Rev. D 58, 024017 (1998). arXiv:gr-qc/9712041

    Article  MathSciNet  ADS  Google Scholar 

  45. S. Hod, T. Piran, Phys. Rev. D 58, 024018 (1998). arXiv:gr-qc/9801001

    Article  MathSciNet  ADS  Google Scholar 

  46. S. Hod, T. Piran, Phys. Rev. D 58, 044018 (1998). arXiv:gr-qc/9801059

    Article  ADS  Google Scholar 

  47. S. Hod, T. Piran, Phys. Rev. D 58, 024019 (1998). arXiv:gr-qc/9801060

    Article  MathSciNet  ADS  Google Scholar 

  48. S. Hod, Phys. Rev. D 58, 104022 (1998). arXiv:gr-qc/9811032

    Article  MathSciNet  ADS  Google Scholar 

  49. S. Hod, Phys. Rev. D 61, 024033 (2000). arXiv:gr-qc/9902072

    Article  MathSciNet  ADS  Google Scholar 

  50. S. Hod, Phys. Rev. D 61, 064018 (2000). arXiv:gr-qc/9902073

    Article  MathSciNet  ADS  Google Scholar 

  51. L. Barack, Phys. Rev. D 61, 024026 (2000)

    Article  MathSciNet  ADS  Google Scholar 

  52. S. Hod, Phys. Rev. Lett. 84, 10 (2000). arXiv:gr-qc/9907096

    Article  ADS  Google Scholar 

  53. S. Hod, Phys. Rev. D 60, 104053 (1999). arXiv:gr-qc/9907044

    Article  MathSciNet  ADS  Google Scholar 

  54. S. Hod, Class. Quantum Gravity 26, 028001 (2009). arXiv:0902.0237

    Article  MathSciNet  ADS  Google Scholar 

  55. S. Hod, Class. Quantum Gravity 18, 1311 (2001). arXiv:gr-qc/0008001

    Article  MathSciNet  ADS  MATH  Google Scholar 

  56. S. Hod, Phys. Rev. D 66, 024001 (2002). arXiv:gr-qc/0201017

    Article  MathSciNet  ADS  Google Scholar 

  57. R.J. Gleiser, R.H. Price, J. Pullin, Class. Quantum Gravity 25, 072001 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  58. M. Tiglio, L.E. Kidder, S.A. Teukolsky, Class. Quantum Gravity 25, 105022 (2008)

    Article  ADS  Google Scholar 

  59. R. Moderski, M. Rogatko, Phys. Rev. D 77, 124007 (2008)

    Article  ADS  Google Scholar 

  60. X. He, J. Jing, Nucl. Phys. B 755, 313 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  61. H. Koyama, A. Tomimatsu, Phys. Rev. D 65, 084031 (2002)

    Article  MathSciNet  ADS  Google Scholar 

  62. B. Wang, C. Molina, E. Abdalla, Phys. Rev. D 63, 084001 (2001)

    Article  ADS  Google Scholar 

  63. A. Zenginoglu, M. Tiglio, Phys. Rev. D 80, 024044 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  64. A.J. Amsel, G.T. Horowitz, D. Marolf, M.M. Roberts, J. High Energy Phys. 0909, 044 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  65. Ya.B. Zel’dovich, Pis’ma Zh. Eksp. Teor. Fiz. 14, 270 (1971). JETP Lett. 14, 180 (1971)

    Google Scholar 

  66. Ya.B. Zel’dovich, Zh. Eksp. Teor. Fiz. 62, 2076 (1972). Sov. Phys. JETP, 35, 1085 (1972)

    Google Scholar 

  67. W.H. Press, S.A. Teukolsky, Astrophys. J. 185, 649 (1973)

    Article  MathSciNet  ADS  Google Scholar 

  68. W.H. Press, S.A. Teukolsky, Nature 238, 211 (1972)

    Article  ADS  Google Scholar 

  69. T. Damour, N. Deruelle, R. Ruffini, Lett. Nuovo Cimento 15, 257 (1976)

    Article  ADS  Google Scholar 

  70. T.M. Zouros, D.M. Eardley, Ann. Phys. 118, 139 (1979)

    Article  ADS  Google Scholar 

  71. S. Detweiler, Phys. Rev. D 22, 2323 (1980)

    Article  ADS  Google Scholar 

  72. H. Furuhashi, Y. Nambu, Prog. Theor. Phys. 112, 983 (2004)

    Article  ADS  MATH  Google Scholar 

  73. V. Cardoso, O.J.C. Dias, J.P.S. Lemos, S. Yoshida, Phys. Rev. D 70, 044039 (2004)

    Article  MathSciNet  ADS  Google Scholar 

  74. V. Cardoso, O.J.C. Dias, J.P.S. Lemos, S. Yoshida, Phys. Rev. D 70, 049903 (2004). Erratum

    Article  MathSciNet  ADS  Google Scholar 

  75. V. Cardoso, J.P.S. Lemos, Phys. Lett. B 621, 219 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  76. V. Cardoso, S. Yoshida, J. High Energy Phys. 0507, 009 (2005)

    Article  MathSciNet  ADS  Google Scholar 

  77. H. Witek, V. Cardoso, A. Ishibashi, U. Sperhake. arXiv:1212.0551

  78. S.R. Dolan, Phys. Rev. D 76, 084001 (2007)

    Article  MathSciNet  ADS  Google Scholar 

  79. S. Hod, O. Hod, Phys. Rev. D 81, 061502 (2010). Rapid communication. arXiv:0910.0734

    Article  MathSciNet  ADS  Google Scholar 

  80. S. Hod, O. Hod. arXiv:0912.2761

  81. H.R. Beyer, J. Math. Phys. 52, 102502 (2011)

    Article  MathSciNet  ADS  Google Scholar 

  82. S. Hod, Phys. Lett. B 708, 320 (2012). arXiv:1205.1872

    Article  ADS  Google Scholar 

  83. S. Hod, Phys. Lett. B 713, 505 (2012)

    Article  MathSciNet  ADS  Google Scholar 

  84. S. Hod, Phys. Rev. D 86, 104026 (2012). arXiv:1211.3202

    Article  ADS  Google Scholar 

  85. S. Hod, Phys. Lett. B 718, 1489 (2012)

    MathSciNet  ADS  Google Scholar 

  86. S.R. Dolan. arXiv:1212.1477

  87. S. Chandrasekhar, The Mathematical Theory of Black Holes (Oxford University Press, New York, 1983)

    MATH  Google Scholar 

  88. R.P. Kerr, Phys. Rev. Lett. 11, 237 (1963)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  89. S.A. Teukolsky, Phys. Rev. Lett. 29, 1114 (1972)

    Article  ADS  Google Scholar 

  90. S.A. Teukolsky, Astrophys. J. 185, 635 (1973)

    Article  MathSciNet  ADS  Google Scholar 

  91. A. Ronveaux, Heun’s Differential Equations (Oxford University Press, Oxford, 1995)

    MATH  Google Scholar 

  92. C. Flammer, Spheroidal Wave Functions (Stanford University Press, Stanford, 1957)

    MATH  Google Scholar 

  93. P.P. Fiziev. arXiv:0902.1277

  94. R.S. Borissov, P.P. Fiziev. arXiv:0903.3617

  95. P.P. Fiziev, Phys. Rev. D 80, 124001 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  96. P.P. Fiziev, Class. Quantum Gravity 27, 135001 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  97. M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1970)

    Google Scholar 

  98. T. Hartman, W. Song, A. Strominger, J. High Energy Phys. 1003, 118 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  99. S. Hod, Phys. Rev. Lett. 100, 121101 (2008). arXiv:0805.3873

    Article  MathSciNet  ADS  Google Scholar 

  100. S. Hod, Phys. Lett. B 717, 462 (2012)

    Article  ADS  Google Scholar 

  101. L.E. Simone, C.M. Will, Class. Quantum Gravity 9, 963 (1992)

    Article  MathSciNet  ADS  Google Scholar 

  102. P.M. Morse, H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953)

    MATH  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

  1. The Ruppin Academic Center, Emeq Hefer, 40250, Israel

    Shahar Hod

  2. The Hadassah Institute, Jerusalem, 91010, Israel

    Shahar Hod

Authors
  1. Shahar Hod
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shahar Hod.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hod, S. Stationary resonances of rapidly-rotating Kerr black holes. Eur. Phys. J. C 73, 2378 (2013). https://doi.org/10.1140/epjc/s10052-013-2378-x

Download citation

  • Received:

  • Published:

  • DOI: https://doi.org/10.1140/epjc/s10052-013-2378-x

Keywords

Search

Navigation