Displaced orbits and electric-magnetic black hole binaries

In presence of magnetic fields, the orbits of charged particles can be displaced from the equatorial plane. We study circular orbits of electrically charged massive objects around a magnetic black hole in the probe approximation. We show that there exist a one-parameter family of circular orbits at constant polar angle $\theta$ and constant radius $r$, parameterized by the angular momentum. The angle $\theta$ can be made arbitrarily small by increasing the charge-to-mass ratio of the orbiting particle, but when this is a Reissner-Nordstr\" om black hole, the condition $q\leq m$ implies that orbits exist only for $|\theta-\frac{\pi}{2}|< {\rm arccot}(2\sqrt{2})$. We show that the circular orbits are stable under small perturbations in the $\theta$ and $r$ directions. We also discuss the Newtonian approximation and a binary system of electric and magnetic black holes, each one describing a circular orbit with no central force.