A sufficient condition for the development of superradiant instabilities in charged black-hole spacetimes

Abstract

The physical and mathematical properties of charged black holes that are linearly coupled to charged massive scalar fields are studied analytically. In particular, we prove that, in the eikonal large-mass regime $M\mu\gg1$, the compact dimensionless inequality $\Phi_{\text{H}}>Q/M$ provides a sufficient condition for the development of superradiant instabilities in the curved black-hole spacetime [here $\{M,Q,\Phi_{\text{H}}\}$ are respectively the mass, the electric charge, and the horizon electrostatic potential of the central black hole and $\mu$ is the proper mass of the field]. The familiar charged Reissner-Nordstr\"om black hole does not satisfy this inequality. On the other hand, we explicitly prove that all charged Ay\'on-Beato-Garc\'ia (ABG) black-hole spacetimes satisfy this analytically derived sufficient condition and may therefore become superradiantly unstable to perturbations of charged massive scalar fields.