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 super-radiant scattering. This scattering process in the super-radiant regime omega < qQ/r(+) (here omega, 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 analog 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) equivalent to (r(m) - r(+))/r(+) << tau, where tau equivalent to (r(+) - r(-))/r(+) is the dimensionless temperature of the black hole. In particular, we derive analytic expressions for the oscillation frequencies R omega and the instability growth time scales 1/T omega of the super-radiant confined fields. Remarkably, we find a simple linear scaling T omega proportional to qQ/r(+) for the imaginary part of the resonances in the asymptotic qQ >> (tau/x(m))(2) >> 1 regime, which implies that the instability time scale 1/T omega of the system can be made arbitrarily short in the qQ -> infinity limit. The short instability time scale 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 nonlinear end-state of super-radiant instabilities.