Spontaneous scalarization of charged Reissner-Nordstr\"om black holes: Analytic treatment along the existence line

TL;DR

This paper analytically derives the discrete resonant spectrum of scalar fields supported by charged Reissner-Nordström black holes, revealing the properties of scalarization bands and confirming results with numerical methods.

Contribution

It provides a compact analytical formula for the scalar field spectrum along the existence line, advancing understanding of black hole scalarization phenomena.

Findings

Derived a compact formula for the scalar field resonant spectrum.

Confirmed analytical results with numerical computations.

Characterized the scalarization bands and their dependence on charge.

Abstract

It has recently been demonstrated that charged black holes can support spatially regular matter configurations made of massless scalar fields which are non-minimally coupled to the electromagnetic field of the charged spacetime. Intriguingly, using numerical techniques, it has been revealed that the resonant spectra of the composed charged-black-hole-nonminimally-coupled-scalar-field configurations are characterized by charge-dependent discrete scalarization bands $\alpha\in\{[\alpha^{-}_{n}({\bar Q}),\alpha^{+}_{n}({\bar Q}]\}_{n=0}^{n=\infty}$, where $\alpha$ is the dimensionless coupling constant of the theory and ${\bar Q}\equiv Q/M$ is the dimensionless charge-to-mass ratio of the central supporting black hole. In the present paper we use {\it analytical} techniques in order to study the physical and mathematical properties of the spatially regular non-minimally coupled scalar field configurations (linearized scalar clouds) which are supported by the central charged Reissner-Nordstr\"om black holes. In particular, we derive a remarkably compact formula for the discrete resonant spectrum $\{\alpha^-_n({\bar Q})\}_{n=0}^{n=\infty}$ which characterizes the composed black-hole-linearized-field configurations along the {\it existence-line} of the system, the critical line which separates bare Reissner-Nordstr\"om black holes from hairy scalarized black-hole configurations. The analytical results are confirmed by direct numerical computations.