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

A charged scalar field impinging upon a charged Reissner--Nordstr\"om 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 ω<qQ/r₊ (here ω, 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 qQ1 and for mirror radii rₘ in the near-horizon region xₘ≡(rₘ-r₊)/r₊τ, where τ≡(r₊-r_-)/r₊ is the dimensionless temperature of the black hole. In particular, we derive analytic expressions for the oscillation frequencies Rω and the instability growth time scales 1/Iω of the super-radiant confined fields. Remarkably, we find a simple linear scaling Iω∝qQ/r₊ for the imaginary part of the resonances in the asymptotic qQ(τ/xₘ)²1 regime, which implies that the instability time scale 1/Iω of the system can be made arbitrarily short in the qQ→∞ 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.