Highly damped quasinormal modes of Kerr black holes

Abstract

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.

Highly damped quasinormal modes of Kerr black holes: A complete numerical investigation

“…This effort is in direct proportion to the difficulty of the problem. All previous attempts [13,16,17,18] at probing the asymptotic QNMs of Kerr BHs have been unsuccessful, or at least unsatisfactory. There have been several contradictory "analytical" results, which were either based on incorrect assumptions, or could not probe the highly damped regime [16].…”

“…There have been several contradictory "analytical" results, which were either based on incorrect assumptions, or could not probe the highly damped regime [16]. The few numerical results [13,17,18] are not decisive either, although they definitely show some trend. A numerical investigation is necessary both as a benchmark and as a guide to analytical approaches.…”

“…We improve on previous results by going further in overtone number than ever before, in order to really probe the asymptotic regime. The starting point for our analysis is, as previously [13,18], Leaver's continued fraction technique [19] as improved by Nollert [9], with a few appropriate modifications [13]. However, we now decouple the angular and radial equations.…”

Physical Interpretation of the Spectrum of Black Hole Quasinormal Modes

“…0 and n ! 1 [7,[15][16][17]. The result (11) can therefore be consistently taken as an indication of a quantization of the area of the horizon of a Schwarzschild BH.…”

“…0, ! R does not reduce to ln3=8M, but rather vanishes as a 1=3 [7,[15][16][17]. This means that the area quantum becomes arbitrarily small if we give to the BH an infinitesimal rotation.…”