Multipole Moments. II. Curved Space

Abstract

Multipole moments are defined for static, asymptotically flat, source-free solutions of Einstein's equations. The definition is completely coordinate independent. We take one of the 3-surfaces V, orthogonal to the timelike Killing vector, and add to it a single point Λ at infinity. The resulting space inherits a conformal structure from V. The multipole moments of the solution emerge as a collection of totally symmetric, trace-free tensors P, Pa, Pab, ⋯ at Λ. These tensors are obtained as certain combinations of the derivatives of the norm of the timelike Killing vector. (For static space-times, this norm plays the role of a ``Newtonian gravitational potential.'') The formalism is shown to yield the usual multipole moments for a solution of Laplace's equation in flat space, the dependence of these moments on the choice of origin being reflected in the conformal behavior of the P's. As an example, the moments of the Weyl solutions are discussed.