Distorted black holes of the Einstein-Klein-Gordon system

We present two families of solutions $(g,\Phi )$ of the Einstein-Klein-Gordon system minimally coupled to gravity, massless scalar field equations possessing a regular $\mathbb{R}\times S^2$ bifurcating Killing horizon. The solutions may be interpreted as describing the metric $g$ and the field $\Phi$ in an open vicinity containing the event horizon of a Schwarzschild black hole interacting with an exterior, static-axisymmetric distribution of matter carrying scalar charges generating $\Phi$. One family of the solution possesses a spherical bifurcation two sphere and may be interpreted as generated by a distribution of matter carrying scalar charges but of negligible self-gravity. The other family describes an axisymmetric horizon with the distortion caused by the self-gravity of the external matter. For both classes of solutions the scalar field $\Phi$ relaxes into a configuration so that its energy-momentum tensor $T^{\mu \nu }$ on the Killing horizon obeys the property: $T^{\mu \nu }{l}_{\mu }{l}_{\nu }=0$ where $l^{\mu }$ is a null vector normal to the horizon. It is also indicated that the Einstein-Klein-Gordon system ought to admit distorted black holes characterized by an $\mathbb{R}\times S^1\times S^1$ event horizon.