Semi-analytic results for quasi-normal frequencies



The last decade has seen considerable interest in the quasi-normal frequencies [QNFs] of black holes (and even wormholes), both asymptotically flat and with cosmological horizons. There is wide agreement that the QNFs are often of the form ω n = (offset) + in (gap), though some authors have encountered situations where this behaviour seems to fail. To get a better understanding of the general situation we consider a semi-analytic model based on a piecewise Eckart (Pöschl-Teller) potential, allowing for different heights and different rates of exponential fall off in the two asymptotic directions. This model is sufficiently general to capture and display key features of the black hole QNFs while simultaneously being analytically tractable, at least for asymptotically large imaginary parts of the QNFs.

We shall derive an appropriate “quantization condition” for the asymptotic QNFs, and extract as much analytic information as possible. In particular, we shall explicitly verify that the (offset) + in (gap) behaviour is common but not universal, with this behaviour failing unless the ratio of rates of exponential falloff on the two sides of the potential is a rational number. (This is “common but not universal” in the sense that the rational numbers are dense in the reals.) We argue that this behaviour is likely to persist for black holes with cosmological horizons.


Black Holes Classical Theories of Gravity Models of Quantum Gravity 


  1. [1]
    S. Chandrasekhar and S.L. Detweiler, The quasi-normal modes of the Schwarzschild black hole, Proc. Roy. Soc. Lond. A 344 (1975) 441 [SPIRES]. ADSGoogle Scholar
  2. [2]
    K.D. Kokkotas and B.G. Schmidt, Quasi-normal modes of stars and black holes, Living Rev. Rel. 2 (1999) 2 [gr-qc/9909058] [SPIRES].MathSciNetGoogle Scholar
  3. [3]
    H.P. Nollert, Quasinormal modes: the characteristic ‘sound’ of black holes and neutron stars, Class. Quant. Grav. 16 (1999) R159 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  4. [4]
    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] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  5. [5]
    O. Dreyer, Quasinormal modes, the area spectrum and black hole entropy, Phys. Rev. Lett. 90 (2003) 081301 [gr-qc/0211076] [SPIRES].CrossRefADSGoogle Scholar
  6. [6]
    J. Natario and R. Schiappa, On the classification of asymptotic quasinormal frequencies for d-dimensional black holes and quantum gravity, Adv. Theor. Math. Phys. 8 (2004) 1001 [hep-th/0411267] [SPIRES].MATHMathSciNetGoogle Scholar
  7. [7]
    C. Eckart, The penetration of a potential barrier by electrons, Phys. Rev. 35 (1930) 1303 [SPIRES]. CrossRefADSGoogle Scholar
  8. [8]
    G. Pöschl and E. Teller, Bemerkungen zur quantenmechanik des anharmonischen, Z. Phys. 83 (1933) 143. MATHCrossRefADSGoogle Scholar
  9. [9]
    P.M. Morse and H. Feshbach, Methods of theoretical physics, McGraw-Hill, New York U.S.A. (1953).MATHGoogle Scholar
  10. [10]
    P. Boonserm, Rigorous bounds on transmission, reflection, and bogoliubov coefficients, P h.D. Thesis, Victoria University of Wellington, Wellington New Zeland (2009) [math-ph/0907.0045].
  11. [11]
    P. Boonserm amd M. Visser, Transmission resonances, quasi-normal modes and quasi-normal frequencies: key analytic results, arXiv:1005.4483.
  12. [12]
    V. Ferrari and B. Mashhoon, New approach to the quasinormal modes of a black hole, Phys. Rev. D 30 (1984) 295 [SPIRES].MathSciNetADSGoogle Scholar
  13. [13]
    S. Iyer and C.M. Will, Black hole normal modes: a WKB approach. 1. foundations and application of a higher order wkb analysis of potential barrier scattering, Phys. Rev. D 35 (1987) 3621 [SPIRES].ADSGoogle Scholar
  14. [14]
    C.M. Will and J.W. Guinn, Tuneling near the peaks of potential barriers: consequences of higher-order Wentzel-Kramers-Brillouin corrections, Phys. Rev. A 37 (1988) 3674.ADSGoogle Scholar
  15. [15]
    J.W. Guinn, C.M. Will, Y. Kojima and B.F. Schutz, High overtone normal modes of Schwarzschild black holes, Class. Quant. Grav. 7 (1990) L47 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  16. [16]
    R.A. Konoplya, Quasinormal behavior of the d-dimensional Schwarzschild black hole and higher order WKB approach, Phys. Rev. D 68 (2003) 024018 [gr-qc/0303052] [SPIRES].ADSGoogle Scholar
  17. [17]
    A.J.M. Medved, D. Martin and M. Visser, Dirty black holes: quasinormal modes, Class. Quant. Grav. 21 (2004) 1393 [gr-qc/0310009] [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  18. [18]
    A.J.M. Medved, D. Martin and M. Visser, Dirty black holes: quasinormal modes for squeezed horizons, Class. Quant. Grav. 21 (2004) 2393 [gr-qc/0310097] [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  19. [19]
    T. Padmanabhan, Quasi normal modes: a simple derivation of the level spacing of the frequencies, Class. Quant. Grav. 21 (2004) L1 [gr-qc/0310027] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  20. [20]
    T.R. Choudhury and T. Padmanabhan, Quasi normal modes in Schwarzschild-deSitter spacetime: a simple derivation of the level spacing of the frequencies, Phys. Rev. D 69 (2004) 064033 [gr-qc/0311064] [SPIRES].MathSciNetADSGoogle Scholar
  21. [21]
    H.R. Beyer, On the completeness of the quasinormal modes of the Poeschl-Teller potential, Commun. Math. Phys. 204 (1999) 397 [gr-qc/9803034] [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  22. [22]
    N. Andersson, A numerically accurate investigation of black hole normal modes, P. Roy. Soc. Lond. A. Mat. A 439 (1992) 47.CrossRefADSGoogle Scholar
  23. [23]
    N. Andersson and C.J. Howls, The asymptotic quasinormal mode spectrum of non-rotating black holes, Class. Quant. Grav. 21 (2004) 1623 [gr-qc/0307020] [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  24. [24]
    E.W. Leaver, An Analytic representation for the quasi normal modes of Kerr black holes, Proc. Roy. Soc. Lond. A 402 (1985) 285 [SPIRES].MathSciNetADSGoogle Scholar
  25. [25]
    E.W. Leaver, Spectral decomposition of the perturbation response of the Schwarzschild geometry, Phys. Rev. D 34 (1986) 384 [SPIRES].MathSciNetADSGoogle Scholar
  26. [26]
    E.W. Leaver, Quasinormal modes of Reissner-Nordström black holes, Phys. Rev. D 41 (1990) 2986 [SPIRES].MathSciNetADSGoogle Scholar
  27. [27]
    L. Motl, An analytical computation of asymptotic Schwarzschild quasinormal frequencies, Adv. Theor. Math. Phys. 6 (2003) 1135 [gr-qc/0212096] [SPIRES].MathSciNetGoogle Scholar
  28. [28]
    L. Motl and A. Neitzke, Asymptotic black hole quasinormal frequencies, Adv. Theor. Math. Phys. 7 (2003) 307 [hep-th/0301173] [SPIRES].MathSciNetGoogle Scholar
  29. [29]
    S. Das and S. Shankaranarayanan, High frequency quasi-normal modes for black-holes with generic singularities, Class. Quant. Grav. 22 (2005) L7 [hep-th/0410209] [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  30. [30]
    A. Ghosh, S. Shankaranarayanan and S. Das, High frequency quasi-normal modes for black holes with generic singularities. II: Asymptotically non-flat spacetimes, Class. Quant. Grav. 23 (2006) 1851 [hep-th/0510186] [SPIRES].MATHCrossRefMathSciNetADSGoogle Scholar
  31. [31]
    J.S.F. Chan and R.B. Mann, Scalar wave falloff in asymptotically anti-de Sitter backgrounds, Phys. Rev. D 55 (1997) 7546 [gr-qc/9612026] [SPIRES].ADSGoogle Scholar
  32. [32]
    J.S.F. Chan and R.B. Mann, Scalar wave falloff in topological black hole backgrounds, Phys. Rev. D 59 (1999) 064025 [SPIRES].ADSGoogle Scholar
  33. [33]
    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] [SPIRES].MathSciNetADSGoogle Scholar
  34. [34]
    B. Wang, C.-Y. Lin and E. Abdalla, Quasinormal modes of Reissner-Nordström anti-de Sitter black holes, Phys. Lett. B 481 (2000) 79 [hep-th/0003295] [SPIRES].MathSciNetADSGoogle Scholar
  35. [35]
    B. Wang, C. Molina and E. Abdalla, Evolving of a massless scalar field in Reissner-Nordström Anti-de Sitter spacetimes, Phys. Rev. D 63 (2001) 084001 [hep-th/0005143] [SPIRES].ADSGoogle Scholar
  36. [36]
    J.-M. Zhu, B. Wang and E. Abdalla, Object picture of quasinormal ringing on the background of small Schwarzschild anti-de Sitter black holes, Phys. Rev. D 63 (2001) 124004 [hep-th/0101133] [SPIRES].ADSGoogle Scholar
  37. [37]
    V. Cardoso and J.P.S. Lemos, Quasi-normal modes of Schwarzschild anti-de Sitter black holes: electromagnetic and gravitational perturbations, Phys. Rev. D 64 (2001) 084017 [gr-qc/0105103] [SPIRES].MathSciNetADSGoogle Scholar
  38. [38]
    B. Wang, E. Abdalla and R.B. Mann, Scalar wave propagation in topological black hole backgrounds, Phys. Rev. D 65 (2002) 084006 [hep-th/0107243] [SPIRES].MathSciNetADSGoogle Scholar
  39. [39]
    D.-P. Du, B. Wang and R.-K. Su, Quasinormal modes in pure de Sitter spacetimes, Phys. Rev. D 70 (2004) 064024 [hep-th/0404047] [SPIRES].MathSciNetADSGoogle Scholar
  40. [40]
    B. Wang, C.-Y. Lin and C. Molina, Quasinormal behavior of massless scalar field perturbation in Reissner-Nordström anti-de Sitter spacetimes, Phys. Rev. D 70 (2004) 064025 [hep-th/0407024] [SPIRES].MathSciNetADSGoogle Scholar
  41. [41]
    V. Suneeta, Quasinormal modes for the SdS black hole: an analytical approximation scheme, Phys. Rev. D 68 (2003) 024020 [gr-qc/0303114] [SPIRES].MathSciNetADSGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2010

Authors and Affiliations

  1. 1.School of Mathematics, Statistics, and Operations ResearchVictoria University of WellingtonWellingtonNew Zealand

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