Multipole moments for stationary, non-asymptotically-flat systems in general relativity

A formulation of multipole moments generalizing that of Thorne is proposed for the stationary, vacuum region of spacetime surrounding a source of gravity, without assuming asymptotic flatness. In this formalism, such a region of spacetime is characterized by four sets of moments, the internal mass and current moments (those of the internal source) and the external mass and current moments (those of the external universe), which are read out from a de Donder coordinate expansion of the metric density. These moments uniquely determine the vacuum region of spacetime. The interactions between a gravitating body and an external gravitational field can be described in terms of these moments, in close analogy with Newtonian theory. A derivation, using the vacuum Einstein equation alone, is given of the laws of force and torque for an isolated body acted on by an external field. These laws generalize the results of Thorne and Hartle and of Zhang.