Asymptotic quasinormal modes of Reissner-Nordström and Kerr black holes

TL;DR

This study delivers the first numerical computation of highly damped quasinormal modes for charged (Reissner-Nordström) and rotating (Kerr) black holes, testing long-standing links between classical black hole oscillations and quantum gravity predictions. By extending Leaver’s continued-fraction method and Nollert’s refinements to RN and Kerr, the authors reveal oscillatory, spiraling behavior of QNM frequencies with increasing charge or rotation, and identify asymptotic forms that largely agree with Motl–Neitzke analytic predictions at large overtone index $n$. The results challenge Hod’s conjecture on the real part of asymptotic frequencies and show a robust Kerr-like asymptotic in the $l=m=2$ sector, $ extomega^{Kerr}_{l=m=2}=2\Omega+i2\pi T_H n$, while highlighting the importance of finite-$n$ corrections and potential limits near extremality. Overall, the work strengthens the bridge between classical BH dynamics and quantum gravity considerations, while clarifying the limits of proposed area-quantization links for charged and rotating black holes.

Abstract

According to a recent proposal, the so-called Barbero-Immirzi parameter of Loop Quantum Gravity can be fixed, using Bohr's correspondence principle, from a knowledge of highly-damped black hole oscillation frequencies. Such frequencies are rather difficult to compute, even for Schwarzschild black holes. However, it is now quite likely that they may provide a fundamental link between classical general relativity and quantum theories of gravity. Here we carry out the first numerical computation of very highly damped quasinormal modes (QNM's) for charged and rotating black holes. In the Reissner-Nordström case QNM frequencies and damping times show an oscillatory behaviour as a function of charge. The oscillations become faster as the mode order increases. At fixed mode order, QNM's describe spirals in the complex plane as the charge is increased, tending towards a well defined limit as the hole becomes extremal. Kerr QNM's have a similar oscillatory behaviour when the angular index $m=0$. For $l=m=2$ the real part of Kerr QNM frequencies tends to $2Ω$, $Ω$ being the angular velocity of the black hole horizon, while the asymptotic spacing of the imaginary parts is given by $2πT_H$.

Figures (7)

• Figure 1: The top two panels show the behaviour of the $n=5$ and $n=10$ QNM frequencies in the complex $\omega$ plane. The $n=10$ mode "spirals in" towards its value in the extremal charge limit; the number of spirals described by each mode increases roughly as the mode order $n$. The panels in the second row show how the $n=10$ spiral "unwinds" as the angular index $l$ is increased (in other words, the asymptotic behaviour sets in later for larger $l$'s). Finally, the bottom panel shows a high-$l$ mode trajectory "pointing" to its limit as the charge becomes extremal. In all cases, we have marked by an arrow the frequency corresponding to the Schwarzschild limit ($Q=0$).
• Figure 2: Real part of the RN QNM frequencies as a function of charge for $n=5,~10,~30,~60,~5000,~10000,~100000$. As the mode order increases the computation becomes more and more time consuming, the oscillations become faster, and a good numerical sampling is rather difficult to achieve; therefore in the last plot we use different symbols (small squares, circles and triangles) to display the actually computed points. For $n=5000,~10000,~100000$ we also compare to the prediction of the analytic formula (\ref{['MNf']}) derived by Motl and Neitzke MN. The oscillatory behaviour is reproduced extremely well by their formula, but the disagreement increases for small charge: formula (\ref{['MNf']}) does not yield the correct Schwarzschild limit.
• Figure 3: Imaginary part of the RN QNM frequencies as a function of charge for $n=10,~30,~60,~5000$. For $n=5000$ we also display the actually computed points, and compare to the prediction of the analytic formula (\ref{['MNf']}). As for the real part, the oscillations are reproduced extremely well, but the disagreement with our numerical data increases for small charge.
• Figure 4: Trajectories of Kerr modes having $m=0$ in the complex-$\omega$ plane. The left panel corresponds to $l=3,~n=15$ and the right panel to $l=3,~n=20$. The number of spirals increases with the mode order, as in the RN case. We have marked by an arrow the point in the plane corresponding to the Schwarzschild limit.
• Figure 5: Real part of Kerr modes having $m=0$ as a function of $a$. Labels indicate the corresponding values of $l$ and of the mode order $n$.