Multipole Moments in General Relativity —Static and Stationary Vacuum Solutions—

Abstract

In the first part of this paper, we review the most important properties of the axisymmetric and stationary solutions of Einstein's vacuum field equations, the solution generating techniques developed in the last few years the relativistic definitions of multipole moments, and their use for obtaining and investigating static and stationary axisymmetric vacuum solutions. In the second part, we derive the general static axisymmetric vacuum solution in prolate spheroidal coordinates. Finally, this solution with all mass multipole moments is generalized to include a further parameter related to the rotation of the mass. This stationary solution can be used in order to describe the exterior gravitational field of a rotating deformed mass.