Multipole expansions of the general-relativistic gravitational field of the external universe

The generic, vacuum, dynamical gravitational field in the vicinity of a freely falling observer is expanded in powers of distance away from the observer's spatial origin (i.e., in distance away from his timelike-geodesic world line). The expansion is determined fully, aside from coordinate freedom, by two families of time-dependent multipole moments: ''electric-type moments'' and ''magnetic-type moments'': which characterize the gravitational influence of the external universe. These ''external multipole moments'' are defined covariantly in terms of the Riemann curvature tensor and its spatial derivatives, evaluated on the observer's world line. The properties of these moments are discussed, and an analysis is given of the structure of the gravitational field's multipole expansion for the special case of de Donder coordinates. In de Donder coordinates the expansion involves only integral powers of distance from the origin; no logarithmic terms occur in this multiparameter expansion