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The charged black-hole bomb: A lower bound on the charge-to-mass ratio of the explosive scalar field

The well-known superradiant amplification mechanism allows a charged scalar field of proper mass $μ$ and electric charge $q$ to extract the Coulomb energy of a charged Reissner-Nordström black hole. The rate of energy extraction can grow exponentially in time if the system is placed inside a reflecting cavity which prevents the charged scalar field from escaping to infinity. This composed black-hole-charged-scalar-field-mirror system is known as the {\it charged black-hole bomb}. Previous numerical studies of this composed physical system have shown that, in the linearized regime, the inequality $q/μ>1$ provides a necessary condition for the development of the superradiant instability. In the present paper we use analytical techniques to study the instability properties of the charged black-hole bomb in the regime of linearized scalar fields. In particular, we prove that the lower bound ${{q}\overμ}>\sqrt{{r_{\text{m}/r_--1}\over{r_{\text{m}}/r_+-1}}}$ provides a necessary condition for the development of the superradiant instability in this composed physical system (here $r_{\pm}$ are the horizon radii of the charged Reissner-Nordström black hole and $r_{\text{m}}$ is the radius of the confining mirror). This {\it analytically} derived lower bound on the superradiant instability regime of the composed black-hole-charged-scalar-field-mirror system is shown to agree with direct {\it numerical} computations of the instability spectrum.