Motion of rotating black holes in homogeneous scalar fields: A general case

In this paper we consider the motion of a rotating black hole through a static, homogeneous, massless scalar field. In the general case, a constant vector of the field gradient can be timelike, spacelike, or null. We consider and compare all of these cases. We demonstrate that as a result of the interaction of the black hole with the scalar field, its mass, spin, and relative velocity with respect to the field can change. We obtain the equations describing the evolution of these parameters and present solutions of the obtained equations for some simple cases.