Rapid growth of superradiant instabilities for charged black holes in a cavity

TL;DR

This paper demonstrates that charged black holes in a cavity exhibit superradiant instabilities that grow significantly faster than those in rotating black holes, offering a promising model for studying non-linear development of such instabilities.

Contribution

The study shows that charged black holes with mirror-like boundaries have superradiant instabilities with growth rates much higher than in rotating black holes, supported by analytic approximation.

Findings

Unstable modes in charged black holes grow faster than in rotating cases.

Maximum Im(ω) for charged black holes is about 0.07, much larger than 10^{-5}.

Analytic understanding explains the faster instability timescale in the charged case.

Abstract

Confined scalar fields, either by a mass term or by a mirror-like boundary condition, have unstable modes in the background of a Kerr black hole. Assuming a time dependence as $e^{-i\omega t}$, the growth time-scale of these unstable modes is set by the inverse of the (positive) imaginary part of the frequency, Im$(\omega)$, which reaches a maximum value of the order of Im$(\omega)M\sim 10^{-5}$, attained for a mirror-like boundary condition, where $M$ is the black hole mass. In this paper we study the minimally coupled Klein-Gordon equation for a charged scalar field in the background of a Reissner-Nordstr\"om black hole and show that the unstable modes, due to a mirror-like boundary condition, can grow several orders of magnitude faster than in the rotating case: we have obtained modes with up to Im$(\omega)M\sim 0.07$. We provide an understanding, based on an analytic approximation, to why the instability in the charged case has a smaller timescale than in the rotating case. This faster growth, together with the spherical symmetry, makes the charged case a promising model for studies of the fully non-linear development of superradiant instabilities.