Exact solution for a binary system of unequal counter-rotating black holes

Abstract

A complete solution describing a binary system constituted by two unequal counter-rotating black holes with a massless strut in between is presented. It is expressed in terms of four arbitrary parameters: the half length of the two rods representing the black hole horizons σ1 and σ2, the total mass M and the relative distance R between the centers of the horizons. The explicit parametrization of this solution in terms of physical parameters, i.e., the Komar masses M1 and M2, the Komar angular momenta J1 and J2 (having J1 and J2 opposite signs) and the coordinate distance R, led us to a four-parameter subclass in which the five physical parameters satisfy a simple algebraic relation, which generalizes the two statements made by Bonnor, in order to remove the additional contributions from the massless spinning rods outside the black holes. Moreover, the interaction force turns out to be of the same form as in the double-Schwarzschild static case.