Asymptotic black hole quasinormal frequencies

TL;DR

This work develops a simplified monodromy-based method to obtain the asymptotic, highly damped quasinormal frequencies of black holes. By analytically continuing the Regge-Wheeler-type equation in the complex radial plane and equating horizon monodromies to near-singularity Bessel behavior, it derives a universal condition $e^{\beta\omega}=-(1+2\cos\pi j)$, with $\beta$ tied to the Hawking temperature, yielding $\text{Re}\,\omega = T_{ ext{Hawking}}\log(3)$ for Schwarzschild in $d\ge4$ for scalar and some gravitational perturbations. In four-dimensional Reissner-Nordström, the spectrum becomes generically aperiodic, governed by a modified monodromy relation $e^{\beta\omega}=-(1+2\cos\pi j) - e^{k^2\beta\omega}(2+2\cos\pi j)$. The results illuminate a deep link between horizon thermodynamics, singularity structure, and the asymptotic spectrum, with potential implications for quantum gravity frameworks such as loop quantum gravity.

Abstract

We give a new derivation of the quasinormal frequencies of Schwarzschild black holes in d>=4 and Reissner-Nordstrom black holes in d=4, in the limit of infinite damping. For Schwarzschild in d>=4 we find that the asymptotic real part is T_Hawking.log(3) for scalar perturbations and for some gravitational perturbations; this confirms a result previously obtained by other means in the case d=4. For Reissner-Nordstrom in d=4 we find a specific generally aperiodic behavior for the quasinormal frequencies, both for scalar perturbations and for electromagnetic-gravitational perturbations. The formulae are obtained by studying the monodromy of the perturbation analytically continued to the complex plane; the analysis depends essentially on the behavior of the potential in the "unphysical" region near the black hole singularity.