Motion of a weakly charged rotating black hole in a homogeneous electromagnetic field

In this paper, we consider a rotating black hole with electric and magnetic monopole charges that moves in a static homogeneous electromagnetic field. We assume that both the charges and the fields are weak, so that they have no effect on the spacetime geometry, which is described by the Kerr metric. We present exact solutions to Maxwell's equations describing the field of a charged rotating black hole moving in an external field background. We use these solutions to calculate the energy, momentum, and angular momentum fluxes of the electromagnetic field into the black hole. Using these results, we obtain expressions for the torque and force acting on the moving charged rotating black hole arising as a result of its interaction with the external electromagnetic field. We calculate these quantities both in the frame comoving with the black hole and in the frame of the external background field. We use this result to derive the equations that govern the change in the black hole's mass and spin as well as the motion of the black hole. We provide exact solutions for specific cases which illustrate the role of charge and spin on the motion of the black hole.