On the multipole expansion of the gravitational field

TL;DR

This paper develops a comprehensive multipole expansion framework in general relativity for slowly-moving isolated sources, linking source properties to observable gravitational radiation, aiding future experimental comparisons.

Contribution

It introduces definitions for source multipole moments valid to all orders in post-Newtonian expansion, connecting them to the stress-energy pseudo-tensor and radiative moments.

Findings

Derived expressions for source multipole moments.

Established relation between source and radiative multipole moments.

Facilitated comparison of gravitational wave observations with theory.

Abstract

This paper constructs the multipole expansion (in general relativity) of the gravitational field generated by a slowly-moving isolated source. 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. Previously obtained expressions of the source multipole moments are recovered in the appropriate limits. 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 non-linear functionals of the source moments, the present paper permits relating the radiation field far from a slowly-moving source to the stress-energy pseudo-tensor of the source. This should be useful when comparing to theory the future observations of gravitational radiation by the LIGO and VIRGO experiments.