A class of exact static spherically symmetric solutions of the Einstein–Maxwell gravity coupled to a massless scalar field is obtained in the harmonic coordinates of Minkowski space-time. For each value of the coupling constant a, these solutions are characterized by a set of three parameters, the physical mass μ0, the electric charge Q0 and the scalar-field parameter k. We find that the solutions for both gravitational and electromagnetic fields are not only affected by the scalar field, but also the nontrivial coupling with matter constrains the scalar field itself. In particular, we find that the constant k differs generically from ±1/2, falling into the interval [Formula: see text]. It takes these values only for black holes or in the case when a scalar field [Formula: see text] is totally decoupled from the matter. Our results differ from those previously obtained in that the presence of an arbitrary coupling constant a gives an opportunity to rule out the nonphysical horizons. In one of the special cases, the obtained solution corresponds to a charged dilatonic black hole with only one horizon μ+ and hence to the Kaluza–Klein case. The most remarkable property of this result is that the metric, the scalar curvature, and both the 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 resulted in a contraction of the horizon. The behavior of the scalar curvature was analysed.
Correction to: Acoustic black holes: massless scalar field analytic solutions and analogue Hawking radiation
