Long-lived resonances of massive scalar fields in the Reissner-Nordström black-hole spacetime: Analytic treatment in the large-mass regime

TL;DR

This work analytically characterizes long-lived quasi-resonances of a minimally coupled massive scalar field in Reissner-Nordström spacetimes in the large-mass, large-angular-momentum regime. Using WKB techniques on the Schrödinger-like radial equation with potential $V(r;M,Q,\mu,l)$, it derives closed-form spectra for Schwarzschild ($Q=0$) and extremal RN ($Q=M$) and a compact analytic expression for the general charge case ${\bar Q}=Q/M$, yielding the critical mass spectrum ${M\mu_{\text{crit}}({\bar Q},l)}$ via a root ${\bar r}_{\text{crit}}({\bar Q})$ of a cubic. The results show that the imaginary part of the frequency vanishes as ${M\mu}$ approaches ${M\mu_{\text{crit}}({\bar Q},l)}$, indicating arbitrarily long-lived resonances, and establish the regime of validity of the WKB approximation with explicit corrections. These findings provide a first analytic closure to the problem of long-lived RN-massive-scalar resonances and quantify relaxation timescales across the full range of black-hole charge.

Abstract

The physical and mathematical properties of the composed Reissner-Nordström-black-hole-massive-scalar-field system are studied {\it analytically} in the dimensionless large-mass $Mμ\gg1$ regime [here $\{M,μ\}$ are respectively the mass of the central black hole and the proper mass of the scalar field]. It is proved that, for a given value ${\bar Q}\equiv Q/M$ of the dimensionless charge parameter of the central black hole, the system is characterized by the presence of quasi-resonances, linearized perturbation modes with arbitrarily long lifetimes. In particular, using analytical techniques, we determine the black-hole-field critical mass spectrum $\{Mμ_{\text{crit}}({\bar Q})\}$ which characterizes the long-lived resonances of the composed physical system.