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

TL;DR

This work analyzes superradiant instabilities of a massive, charged scalar field in a Reissner-Nordström black hole placed inside a cavity with a mirror boundary. Using a mirrored quasi-bound-state framework and a Schrödinger-like radial equation, it finds growth rates up to $\mathrm{Im}(\omega) M \sim 0.07$, far exceeding the Kerr mirror-case. An analytic near-limit expression with $\mathrm{Re}(\omega_n) \approx j_{\ell+1/2,n}/r_m$ and $\mathrm{Im}(\omega_n) \approx -\gamma\, r_m^{-2(\ell+1)}(\mathrm{Re}(\omega_n)-\omega_c)$, where $\omega_c = q\,\Phi_+$, clarifies how increasing the charge $q$ (and thus the critical frequency) boosts the instability. The authors argue the nonlinear endpoint is a scalar condensate (hair) around the charged black hole, and advocate the RN-in-a-cavity setup as a practical, highly symmetric model to study the fully nonlinear development of superradiant instabilities.

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 $e^{-iωt}$, the growth time-scale of these unstable modes is set by the inverse of the (positive) imaginary part of the frequency, Im$(ω)$, which reaches a maximum value of the order of Im$(ω)M\sim 10^{-5}$, 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-Nordström 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$(ω)M\sim 0.07$. 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.

Figures (4)

• Figure 1: The imaginary part of the frequency plotted versus the radius of the mirror for various ratios of the scalar charge, $q$ to scalar mass $\mu$. The scalar mass is $\mu=0.3$. We included the line Im$(\omega)=0$ to help visualizing where the curves have Im$(\omega)>0$.
• Figure 2: The imaginary and real part of $\omega$ drawn as a function of the mirror radius $r_m$ for various values of the black hole charge, $Q$, and the scalar mass, $\mu$: $\mu=0.1,0.2,0.3$ for the left, middle and right column respectively. We took $q=0.6$.
• Figure 3: The imaginary and real part of $\omega$ drawn as a function of the mirror radius $r_m$ for various values of the black hole charge, $Q$, and the scalar charge, $q$: $q=0.9,1.5,2.0$ for the left, middle and right column respectively. We took $\mu=0.1$.
• Figure 4: The imaginary part (real part in the inset) of the frequency as a function of $q$ for $Q=0.9$, $\mu=0.1$ and $r_m=5,10,20,50$.