Analytic treatment of the charged black-hole-mirror bomb in the highly explosive regime

A charged scalar field impinging upon a charged Reissner-Nordstrom black hole can be amplified as it scatters off the hole, a phenomenon known as superradiant scattering. This scattering process in the superradiant regime w<qQ/r_+ (here w, q, Q, and r_{+-} are the conserved frequency of the wave, the charge coupling constant of the field, the electric charge of the black hole, and the horizon radii of the black hole, respectively) results in the extraction of Coulomb energy and electric charge from the charged black hole. The black-hole-field system can be made unstable by placing a reflecting mirror around the black hole which prevents the amplified field from escaping to infinity. This charged black-hole-mirror system is the spherically symmetric analogue of the rotating black-hole-mirror bomb of Press and Teukolsky. In the present paper we study analytically the charged black-hole-mirror bomb in the asymptotic regime qQ>>1 and for mirror radii r_m in the near-horizon region x_m=(r_m-r_+)/r_+<<tau, where tau=(r_+-r_-)/r_+ is the dimensionless temperature of the black hole. In particular, we derive analytic expressions for the oscillation frequencies w_R and the instability growth timescales 1/w_I of the superradiant confined fields. Remarkably, we find a simple linear scaling w_I \propto qQ/r_+ for the imaginary part of the resonances in the asymptotic qQ>>(tau/x_m)^2>>1 regime, which implies that the instability timescale 1/w_I of the system can be made arbitrarily short in the qQ-->infinity limit. The short instability timescale found in the linear regime along with the spherical symmetry of the system, make the charged bomb a convenient toy model for future numerical studies aimed to investigate the non-linear end-state of superradiant instabilities.