Two mass solutions of Einstein's vacuum equations: The double Kerr solution

Abstract

The newly developed transformations which allow one to generate solutions of Einstein's vacuum equations from given ones have become in the meantime very powerful tools for constructing new solutions. We prove the equivalence of Kramer-Neugebauer and HKX transformations and construct the double Kerr solution by means of four HKX transformations. The resulting solution is analysed in detail. We employ an analytic continuation of the parameters to pass from underextreme to hyperextreme constituents and pose conditions for asymptotic flatness, for existence of an axis between the massive objects, and for balance caused by the repulsive interaction of the angular momenta. We define mass and angular momentum of the single constituents and compute these quantities explicitly in addition to the total mass and the total angular momentum of the solution. The distance and force between the massive objects are defined and given in a suitable approximation. We prove that two black holes—constituents possessing a horizon—cannot be in balance. The above-mentioned conditions are solved for equal hyperextreme constituents which balance due to spin-spin repulsion. We give the distance of balance. Finally, we compare rotating and nonrotating two-mass systems of equal masses and equal distance between them and estimate the change of force between the masses caused by the spin-spin repulsion.