Black hole quasinormal modes: hints of quantum gravity?

TL;DR

This review analyzes the quasinormal mode spectrum of Schwarzschild, RN, and Kerr black holes, detailing both established and new calculations of angular eigenvalues $_sA_{lm}$ and extremal RN modes, using Leaver’s continued-fraction method and its refinements. It highlights the intricate high-damping behavior, including Schwarzschild limits $ ext{Re}( obreak\omega)\to T_H \ln 3$ and Motl–Neitzke asymptotics for RN, while revealing surprising features in charged and rotating spacetimes such as spiral trajectories, isospectralities, and Kerr–Newman challenges. The work presses on the potential links between QNMs and quantum gravity, but also stresses persistent puzzles: nontrivial limits, algebraically special modes, multiplet emergence near AS frequencies, and the dependence on inner/horizon structure in RN/Kerr spacetimes. Overall, the paper underscores the need for deeper understanding of the classical-quantum connections in black hole perturbations and the full implications of highly damped, rotating and charged regimes.

Abstract

This is a short review of the quasinormal mode spectrum of Schwarzschild, Reissner-Nordstrom and Kerr black holes. The summary includes previously unpublished calculations of i) the eigenvalues of spin-weighted spheroidal harmonics, and ii) quasinormal frequencies of extremal Reissner-Nordstrom black holes.

Figures (17)

• Figure 1: Quasinormal frequencies for gravitational perturbations with $l=2$ (blue circles) and $l=3$ (red diamonds). Compare eg. Figure 1 in N. In both cases we mark by an arrow the algebraically special mode, that is given analytically by Equation (\ref{['AlgSp']}); a more extensive discussion of this mode is given in section \ref{['sec:kerras']}. Notice that as the imaginary part of the frequency tends to infinity the real part tends to a finite, $l$-independent limit.
• Figure 2: Trajectory described in the complex--frequency plane by the fundamental RN quasinormal mode as the charge is increased. The solid line corresponds to $l=2$ and $Z_2^-$ (perturbations that reduce to the axial--gravitational Schwarzschild case as $Q\to 0$); the dashed line, to $l=1$ and $Z_1^-$ (purely electromagnetic Schwarzschild perturbations in the limit $Q\to 0$). The modes coalesce in the extremal limit $Q\to 1/2$.
• Figure 3: The top two panels show the behavior of the $n=5$ and $n=10$ quasinormal 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 behavior sets in later for larger $l$'s). In all cases, we have marked by an arrow the frequency corresponding to the Schwarzschild limit ($Q=0$).
• Figure 4: Real part of the RN quasinormal 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 behavior is reproduced extremely well by their formula, but the disagreement increases for small charge: formula (\ref{['MNf']}) does not yield $T_H \ln 3$ in the Schwarzschild limit.
• Figure 5: Imaginary part of the RN quasinormal 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.