Reissner-Nordström black holes supporting nonminimally coupled massive scalar field configurations
Shahar Hod
Abstract
It has recently been demonstrated that static spatially regular scalar fields, which are nonminimally 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 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 $\ensuremath{\mu}{r}_{+}\ensuremath{\gg}1$ (here, $\ensuremath{\mu}$ 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 ${{\ensuremath{\alpha}}_{n}(\ensuremath{\mu};Q/M){}}_{n=0}^{n=\ensuremath{\infty}}$ which characterizes the dimensionless coupling parameter of the composed black hole--nonminimally 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 ${\ensuremath{\alpha}}_{0}(\ensuremath{\mu})$ determines the critical existence line of the composed black hole--massive field system, a boundary line which separates nonlinearly 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.