Two-body equilibrium configurations involving one extreme black hole: the electrovacuum case

The present paper fills in the final gap in the search and description of different equilibrium states in the two-body systems consisting of one extreme and one non-extreme components. In the ‘extreme-non-extreme’ case of charged spinning masses we obtain, by making use of an appropriate exact solution of the Einstein-Maxwell equations and solving numerically the corresponding balance equations, the first examples of the ‘extreme-hyperextreme’ equilibrium configurations characterized by positive Komar masses of both Kerr–Newman constituents. Furthermore, we demonstrate that equilibrium in the ‘extreme-subextreme’ stationary electrovac systems is also possible, but all particular equilibrium configurations of this type found by us involve negative mass of one of the constituents. In the electrostatic case which admits a purely analytic treatment we give a rigorous proof of the non-existence of the ‘extreme-non-extreme’ equilibrium configurations in the framework of the double-Reissner–Nordstr?m solution. At the same time, the electrostatic equilibrium between an extreme and a subextreme black holes can be achieved in the uniform external field, provided the two constituents form a specific dihole with zero net charge.