Stability of charged scalar hair on a Reissner-Nordstr\"om black hole

Abstract

The Israel-Carter theorem (also known as the "no-hair theorem") puts a restriction on the existence of parameters other than mass, electric charge, and angular momentum of a black hole. In this context, Bekenstein proposed no-hair theorems in various black hole models with neutral and electrically charged scalar fields. In this paper, we take the Einstein-Maxwell-charged scalar model with an electrically charged scalar field gauge-coupled to the Maxwell field surrounding a charged black hole with a static spherically symmetric metric. In particular, we consider a quadratic scalar potential without any higher order terms and we do not impose any restriction on the magnitude of the scalar charge with respect to the black hole charge. With this setting, we ascertain the validity of all energy conditions coupled with the causality condition, suggesting the possibility of existence of charged hairy solutions. Consequently, we obtain, by exact numerical integration, detailed solutions of the field equations that incorporate backreaction on the spacetime due to the presence of the charged scalar field. The solutions exhibit damped oscillatory behaviours for the charged scalar hair. We also find that the electric potential is a monotonic function of the radial coordinate, as required by electrodynamics. In order to ascertain the existence of our charged hairy solutions, we carry out dynamic stability analyses against time-dependant perturbations about the static solutions. For a definite conclusion, we employ two different methodologies. The first methodology involves a Sturm-Liouville equation, whereas the second methodology employs a Schr\"odinger-like equation, for the dynamic perturbations. We find that our solutions are stable against time-dependant perturbations by both methodologies, confirming the existence of the charged hairy solutions.