Reissner-Nordstr\"om black holes supporting non-minimally coupled massive scalar field configurations

TL;DR

This paper analytically investigates non-minimally coupled massive scalar fields around Reissner-Nordström black holes, deriving a compact formula for the resonant spectrum that determines the boundary between hairy and bald black-hole solutions.

Contribution

It provides an analytical formula for the discrete resonant spectrum of scalar clouds around charged black holes in the large-mass regime, advancing understanding of black-hole scalar field configurations.

Findings

Derived a compact analytical formula for the resonant spectrum.

Confirmed analytical results with numerical computations.

Identified the critical existence-line separating hairy and bald black holes.

Abstract

It has recently been demonstrated that static spatially regular scalar fields, which are non-minimally coupled to the electromagnetic field of a charged central black hole, can be supported in the exterior regions of the black-hole spacetime. In the present paper we use {\it analytical} techniques in order to study the physical and mathematical properties of the externally supported linearized scalar field configurations (scalar 'clouds') in the dimensionless large-mass regime $\mu r_+\gg1$ (here $\mu$ and $r_+$ are respectively the proper mass of the supported scalar field and the outer horizon radius of the central supporting black hole). In particular, we derive a remarkably compact analytical formula for the discrete resonant spectrum $\{\alpha_n(\mu;Q/M)\}_{n=0}^{n=\infty}$ which characterizes the dimensionless coupling parameter of the composed black-hole-non-minimally-coupled-linearized-massive-scalar-field configurations. The physical significance of this resonant spectrum stems from the fact that, for a given value of the dimensionless black-hole electric charge $Q/M$, the fundamental (smallest) eigenvalue $\alpha_0(\mu)$ determines the critical existence-line of the composed black-hole-massive-field system, a boundary line which separates non-linearly coupled hairy charged-black-hole-massive-scalar-field configurations from bald Reissner-Nordstr\"om black holes. The analytical results derived in this paper are confirmed by direct numerical computations.