Black holes with regular horizons in Maxwell-scalar gravity

Abstract

A class of exact static spherically symmetric solutions of the Einstein-Maxwell gravity coupled to a massless scalar field has been obtained in harmonic coordinates of the Minkowski space-time. For each value of the coupling constant $a$, these solutions are characterized by a set of three parameters, the physical mass $μ_0$, the electric charge $Q_0$ and the scalar field parameter $k$. We have found that the solutions for both gravitational and electromagneticfields are not only affected by the scalar field, but also the non-trivial coupling with matter constrains the scalar field itself. In particular, we have found that the constant $k$ generically differs from $\pm 1/2$, falling into the interval $|k|\in [0, {1\over2}\sqrt{1+a^2} \hskip 2pt ]$. It takes these values only for black holes or in the case when a scalar field $ϕ$ is totally decoupled from the matter. Our results differ from those previously obtained in that the presence of arbitrary coupling constant $a$ gives an opportunity to rule out the non-physica horizons. In one of the special cases, the obtained solution corresponds to a charged dilatonic black hole with only one horizon $μ_+$ and hence for the Kaluza-Klein case. The most remarkable property of this result is that the metric, the scalar curvature, and both electromagnetic and scalar fields are all regular on this surface. Moreover, while studying the dilaton charge, we found that the inclusion of the scalar field in the theory result in a contraction of the horizon. The behavior of the scalar curvature was analyzed.