Stability of highly-charged Reissner-Nordström black holes to charged scalar perturbations

The stability of Reissner-Nordström black holes under the influence of neutral perturbation fields was proved by Moncrief four decades ago. However, the superradiant scattering phenomenon, which characterizes the dynamics of charged bosonic fields in these charged black-hole spacetimes, imposes a greater and nontrivial threat on their stability. According to this well-known phenomenon, integer-spin charged fields interacting with a Reissner-Nordström black hole can be amplified (gain energy) by extracting some of the black-hole Coulomb energy. If, in addition to being electrically charged, the incident bosonic fields also possess nonzero rest masses, then the mutual gravitational attraction between the central black hole and the fields may prevent the extracted energy and electric charge from escaping to infinity. One may suspect that the physical mechanism of superradiant amplification of charged bosonic fields in the charged Reissner-Nordström black-hole spacetime, when combined with the confinement mechanism provided by the mutual gravitational attraction between the black hole and the massive fields, may lead to a superradiant instability of the Reissner-Nordström black-hole spacetime. (This suspicion is mainly based on our experience with rotating Kerr black holes, which are known to be characterized by an analogous superradiant instability when coupled to massive bosonic fields.) However, in this paper we show that, for highly-charged Reissner-Nordström black holes in the charge interval $8/9<{(Q/M)}^2<1$, the two physical mechanisms which are required in order to trigger the superradiant instability phenomenon in the black-hole spacetime [namely: (1) the superradiant amplification of incident charged scalar fields by the charged black hole, and (2) the existence of a binding potential well in the black-hole exterior region which prevents the extracted energy and electric charge from escaping to infinity] cannot operate simultaneously. In particular, we shall prove that, in the superradiant regime, there is no trapping potential well in the black-hole exterior region. Our stability results for the highly-charged Reissner-Nordström black holes in the regime $8/9<(Q/M{)}^2<1$, combined with former analytical studies which explored the stability of these charged black holes in the complementary regime ${(Q/M)}^2\le 8/9$, establish the fact that charged Reissner-Nordström black holes are stable to charged massive scalar perturbations in the entire parameter space.