Highly damped quasinormal modes of Kerr black holes

TL;DR

This work systematically computes highly damped Kerr QNM frequencies for scalar, electromagnetic, and gravitational perturbations using Leaver's continued-fraction method, verified by two independent codes. It finds that the common Hod conjecture does not hold universally; notably, for $l=m=2$ gravitational modes the real part tends to $\textomega_R\to 2\Omega$, while many other $m>0$ modes approach $\textomega_R\to m$ in the extremal limit and $m<0$ modes tend to a universal $\textomega_R\approx -m\varpi$ with $\varpi\approx 0.12$. The imaginary-part spacing for $m>0$ modes is universally $2\pi T_H$, tying the spectrum to the black hole temperature, and spiral trajectories characterize $m=0$ modes. The study also uncovers Kerr multiplets near the Schwarzschild algebraically special frequencies, confirming conjectures about algebraically special mode branching but highlighting discrepancies with some analytic predictions, which motivates further analytic work on Kerr QNM asymptotics and potential links to quantum gravity.

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 do not limit our attention to gravitational modes, thus filling some gaps in the existing literature. The frequency of gravitational modes with l=m=2 tends to ω_R=2 Ω, Ωbeing the angular velocity of the black hole horizon. 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 ω_R\simeq -m\varpi, \varpi\simeq 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-Nordstrom (RN) black holes. Finally, for any perturbing field, the asymptotic separation in the imaginary part of consecutive modes with m>0 is given by 2πT_H (T_H 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 (ω_R=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.

Figures (11)

• Figure 1: Each different symbol corresponds to the (numerically computed) value of $\omega_R$ as a function of the mode index $n$, at different selected values of the rotation parameter $a$. The selected values of $a$ are indicated on the right of the plot. Horizontal lines correspond to the predicted asymptotic frequencies $2\Omega$ at the given values of $a$. Convergence to the asymptotic value is clearly faster for larger $a$. In the range of $n$ allowed by our numerical method ($n\lesssim 50$) convergence is not yet achieved for $a\lesssim 0.1$.
• Figure 2: Relative difference between various fit functions and numerical results for the mode with overtone index $n=40$. From top to bottom in the legend, the lines correspond to the relative errors for formulas (\ref{['Hod']}), (\ref{['mOm']}), (\ref{['Om2']}) and (\ref{['mOm2']}).
• Figure 3: Real part of the frequency for different modes with $l=2$ and $m>0$. In both panels we overplot (bold solid line) the prediction of formula (\ref{['mOm']}). The left panel shows the excellent agreement between modes with $l=m=2$ and the asymptotic formula. The right panel shows the different behaviour of modes with $m=1$; these modes have a frequency that "bends" downwards as $n$ increases, showing a local minimum as a function of $a$. In both cases, $\omega_R\to m$ in the extremal limit $a\to 1/2$.
• Figure 4: Real part of the first few modes with $l=2$ and $m<0$. Modes with $m=-1$ are shown in the left panel, modes with $m=-2$ in the right panel. As the mode order $n$ increases, $\omega_R$ seems to approach a (roughly) constant value $\omega_R=-m \varpi$, where $\varpi\simeq 0.12$. Convergence to this limiting value is faster for large values of the rotation parameter $a$ (compare figure \ref{['fig1']}).
• Figure 5: Real parts of some modes with $l=3$ and different values of $m$ (indicated in the plots). When $m>0$, the observed behaviour is reminiscent of modes with $l=2$, $m=1$ (see figure \ref{['fig3']}). Modes with $m<0$ approach a (roughly) constant value $\omega_R=-m\varpi$ (we only show modes with $m=-1$), as they do for $l=2$ (see figure \ref{['fig4']}).