Displaced orbits and electric-magnetic black hole binaries

Abstract: 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.