Black-hole normal modes: A WKB approach. IV. Kerr black holes

Seidel, Edward; Iyer, Sai

doi:10.1103/physrevd.41.374

Cited by 144 publications

(112 citation statements)

References 29 publications

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“…In this paper, we extended the results of several earlier works [9,29,40,41] to compute the quasinormal-mode frequencies and wave functions of a Kerr black hole of arbitrary astrophysical spins, in the eikonal limit (l ) 1). We focused on developing a greater intuitive understanding of their behavior, but in the process, we calculated expressions for large-l quasinormal-mode frequencies that are reasonably accurate even at low l. Specifically, we applied a WKB analysis to the system of equations defined by the angular and radial Teukolsky equations.…”

Section: Conclusion and Discussionmentioning

confidence: 91%

“…Although there has been work by Kokkotas [40] and Seidel and Iyer [41] using WKB methods to compute QNM frequencies of rotating black holes, their results were limited to slowly rotating black holes, because they performed an expansion of the angular separation constant, A lm , for small, dimensionless spin parameters, a=M, and only applied the WKB method to the radial Teukolsky equation to solve for the frequency. In a different approach, Dolan developed a matched-expansion formalism for Kerr black holes of arbitrary spins that can be applied to compute the frequency of QNMs, but only for modes with l ¼ jmj and m ¼ 0 [29].…”

Section: B Methods and Results Of This Articlementioning

confidence: 99%

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Quasinormal-mode spectrum of Kerr black holes and its geometric interpretation

et al. 2012

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There is a well-known, intuitive geometric correspondence between high-frequency quasinormal modes of Schwarzschild black holes and null geodesics that reside on the light ring (often called spherical photon orbits): the real part of the mode's frequency relates to the geodesic's orbital frequency, and the imaginary part of the frequency corresponds to the Lyapunov exponent of the orbit. For slowly rotating black holes, the quasinormal mode's real frequency is a linear combination of the orbit's precessional and orbital frequencies, but the correspondence is otherwise unchanged. In this paper, we find a relationship between the quasinormal-mode frequencies of Kerr black holes of arbitrary (astrophysical) spins and general spherical photon orbits, which is analogous to the relationship for slowly rotating holes. To derive this result, we first use the Wentzel-Kramers-Brillouin approximation to compute accurate algebraic expressions for large-l quasinormal-mode frequencies. Comparing our Wentzel-Kramers-Brillouin calculation to the leading-order, geometric-optics approximation to scalar-wave propagation in the Kerr spacetime, we then draw a correspondence between the real parts of the parameters of a quasinormal mode and the conserved quantities of spherical photon orbits. At next-to-leading order in this comparison, we relate the imaginary parts of the quasinormal-mode parameters to coefficients that modify the amplitude of the scalar wave. With this correspondence, we find a geometric interpretation of two features of the quasinormal-mode spectrum of Kerr black holes: First, for Kerr holes rotating near the maximal rate, a large number of modes have nearly zero damping; we connect this characteristic to the fact that a large number of spherical photon orbits approach the horizon in this limit. Second, for black holes of any spins, the frequencies of specific sets of modes are degenerate; we find that this feature arises when the spherical photon orbits corresponding to these modes form closed (as opposed to ergodically winding) curves.

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Section: Conclusion and Discussionmentioning

confidence: 91%

Section: B Methods and Results Of This Articlementioning

confidence: 99%

Quasinormal-mode spectrum of Kerr black holes and its geometric interpretation

et al. 2012

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“…In turn, these convergence conditions can be expressed as two equations involving continued fractions. Finding QNM frequencies is a two-step procedure: for assigned values of a, ℓ, m and ω, first find the angular separation constant A lm (ω) looking for zeros of the angular continued fraction equation; then replace the corresponding eigenvalue into the radial continued fraction equation, and look for its zeros as a function of ω. Leaver's method is relatively well convergent and numerically stable for highly damped modes, when compared to other techniques [34]. We mention that an alternative, approximate method for finding Kerr quasinormal frequencies has recently been presented [35], which has the advantage of highlighting some physical features of the problem.…”

Section: Methodsmentioning

confidence: 99%

Highly damped quasinormal modes of Kerr black holes

Cardoso²,

et al. 2003

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Motivated by recent suggestions that highly damped black hole quasinormal modes (QNM's) may provide a link between classical general relativity and quantum gravity, we present an extensive computation of highly damped QNM's of Kerr black holes. We perform the computation using two independent numerical codes based on Leaver's continued fraction method. We do not limit our attention to gravitational modes, thus filling some gaps in the existing literature. As already observed in [19], the frequency of gravitational modes with l = m = 2 tends to ωR = 2Ω, Ω being the angular velocity of the black hole horizon. We show that, if Hod's conjecture is valid, this asymptotic behaviour is related to reversible black hole transformations. Other highly damped modes with m > 0 that we computed do not show a similar behaviour. The real part of modes with l = 2 and m < 0 seems to asymptotically approach a constant value ωR ≃ −m̟, ̟ ≃ 0.12 being (almost) independent of a. For any perturbing field, trajectories in the complex plane of QNM's with m = 0 show a spiralling behaviour, similar to the one observed for Reissner-Nordström (RN) black holes. Finally, for any perturbing field, the asymptotic separation in the imaginary part of consecutive modes with m > 0 is given by 2πTH (TH being the black hole temperature). We conjecture that for all values of l and m > 0 there is an infinity of modes tending to the critical frequency for superradiance (ωR = m) in the extremal limit. Finally, we study in some detail modes branching off the so-called "algebraically special frequency" of Schwarzschild black holes. For the first time we find numerically that QNM multiplets emerge from the algebraically special Schwarzschild modes, confirming a recent speculation.

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“…There is a vast literature on Kerr QNMs [57,58,59,60], but the QNM frequencies which are relevant for detection have never been systematically tabulated. Here we list for reference the values of the complex frequencies and separation constants for selected QNMs.…”

Section: Appendix E: Quasinormal Frequencies For Rotating Black Holesmentioning

confidence: 99%

Gravitational-wave spectroscopy of massive black holes with the space interferometer LISA

Berti¹,

Cardoso²,

Will³

2006

Phys. Rev. D

769

1,048

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Newly formed black holes are expected to emit characteristic radiation in the form of quasi-normal modes, called ringdown waves, with discrete frequencies. LISA should be able to detect the ringdown waves emitted by oscillating supermassive black holes throughout the observable Universe. We develop a multi-mode formalism, applicable to any interferometric detectors, for detecting ringdown signals, for estimating black hole parameters from those signals, and for testing the no-hair theorem of general relativity. Focusing on LISA, we use current models of its sensitivity to compute the expected signal-to-noise ratio for ringdown events, the relative parameter estimation accuracy, and the resolvability of different modes. We also discuss the extent to which uncertainties on physical parameters, such as the black hole spin and the energy emitted in each mode, will affect our ability to do black hole spectroscopy.

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Black-hole normal modes: A WKB approach. IV. Kerr black holes

Cited by 144 publications

References 29 publications

Quasinormal-mode spectrum of Kerr black holes and its geometric interpretation

Quasinormal-mode spectrum of Kerr black holes and its geometric interpretation

Highly damped quasinormal modes of Kerr black holes

Gravitational-wave spectroscopy of massive black holes with the space interferometer LISA

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