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No-bomb theorem for charged Reissner-Nordstrom black holes

The fundamental role played by black holes in many areas of physics makes it highly important to explore the nature of their stability. The stability of charged Reissner-Nordström black holes to {\it neutral} (gravitational and electromagnetic) perturbations was established almost four decades ago. However, the stability of these charged black holes under {\it charged} perturbations has remained an open question due to the complexity introduced by the well-known phenomena of superradiant scattering: A charged scalar field impinging on a charged Reissner-Nordström black hole can be {\it amplified} as it scatters off the hole. If the incident field has a non-zero rest mass, then the mass term effectively works as a mirror, preventing the energy extracted from the hole from escaping to infinity. One may suspect that the superradiant amplification of charged fields by the charged black hole may lead to an instability of the Reissner-Nordström spacetime (in as much the same way that rotating Kerr black holes are unstable under rotating scalar perturbations). However, in this Letter we show that, for charged Reissner-Nordström black holes in the regime ${(Q/M)}^2\leq 8/9$, the two conditions which are required in order to trigger a possible superradiant instability [namely: (1) the existence of a trapping potential well outside the black hole, and (2) superradiant amplification of the trapped modes] cannot be satisfied simultaneously. Our results thus support the stability of charged Reissner-Nordström black holes under charged scalar perturbations in the regime ${(Q/M)}^2\leq 8/9$.