On the multipole expansion of the gravitational field

The multipole expansion (in general relativity) of the gravitational field generated by a slowly-moving isolated source is constructed. We introduce some definitions for the source multipole moments, valid to all orders in a post-Newtonian expansion, and depending in a well defined way on the total stress-energy pseudo-tensor of the material and gravitational fields. The source moments parametrize the linearized approximation of the gravitational field exterior to the source, as computed by means of a specific post-Minkowskian algorithm defined in a previous work. Since the radiative multipole moments parametrizing the radiation field far from the source can be obtained as nonlinear functionals of the source moments, the present paper allows one to relate the radiation field far from a slowly-moving source to the stress-energy pseudo-tensor of the source. This should be useful when comparing theory with the future observations of gravitational radiation by the LIGO and VIRGO experiments.