Quasinormal modes of maximally charged black holes

TL;DR

The paper addresses computing quasinormal frequencies $\omega$ for extremal Reissner–Nordström black holes, where the horizon's irregular singular point makes Leaver's continued-fraction method inapplicable in the limit $Q\to M$. It develops a continued-fraction framework by expanding the perturbation about an ordinary point with $u=(r-2)/r$, deriving a five-term recurrence for the coefficients $a_n$, separating into even/odd sequences (with scalar field decoupled) and reducing to three-term recurrences whose $a_{n+1}/a_n$ ratios are fixed by continued fractions. Numerical results for multiple $(s,l)$ modes show agreement with Leaver and with third-order WKB within a few percent, and reveal a striking extremal-limit coincidence: gravitational modes with index $l$ share the same frequencies as electromagnetic modes with index $l-1$. The method provides a robust, broadly applicable tool for extremal and potentially rotating black holes and sheds light on hidden symmetries in the extremal regime with implications for gravitational-wave phenomenology.

Abstract

A new algorithm for computing the accurate values of quasinormal frequencies of extremal Reissner-Nordström black holes is presented. The numerically computed values are consistent with the values earlier obtained by Leaver and those obtained through the WKB method. Our results are more precise than other results known to date. We also find a curious fact that the resonant frequencies of gravitational waves with multi-pole index $l$ coincide with those of electromagnetic waves with multi-pole index $l-1$ in the extremal limit.

Figures (2)

• Figure 1: Quasinormal frequencies for $l=2$ gravitational wave and $l=1,2$ electromagnetic wave are plotted in the $\omega$-plane with the results of Leaver[13]. Each solid line is a trajectory of the first three lowest $l=2$ gravitational mode, parameterized by the charge $Q$. It has a tendency to coincide at the right end with $l=1$ electromagnetic mode which is shown as a small dashed line. Marks from left to right correspond to $Q=0,0.4,0.8,0.9999$ quasinormal frequencies of Leaver. The frequencies of extremal black holes we computed are plotted as diamonds. Dashed lines are $l=2$ electromagnetic modes.
• Figure 2: Solid lines and dashed lines are trajectories of the third order WKB quasinormal frequencies of the gravitational and electromagnetic wave, respectively. Each left endpoint of lines corresponds to the quasinormal frequency of a charged black hole of $Q=0.8$, and each right endpoint corresponds to the frequency in the limit of maximal charge. A trajectory of the gravitational quasinormal frequencies depicted in a solid line meets at the right end with a corresponding dashed line, which is a trajectory of the electromagnetic quasinormal frequencies belonging to lower multi-pole index by one.