On the multipole expansion of the gravitational field

TL;DR

This paper presents a rigorous construction of the gravitational multipole expansion for a slowly moving, isolated source in general relativity, valid to all orders in the post-Newtonian expansion and consistent with the nonlinear MPM framework. It defines explicit source multipole moments via analytic continuation of the stress–energy pseudotensor and derives their relation to the exterior field, including nonlinear corrections that affect radiative moments at infinity. The work provides STF decompositions of the linearized metric in terms of six families of moments, clarifies the link between source moments (I_L,J_L, W_L,X_L,Y_L,Z_L) and radiative moments (U_L,V_L), and discusses tails and gauge effects essential for accurate gravitational-wave modeling. Together with prior and subsequent results, the formalism enables computing radiative signals from compact sources to high post-Newtonian orders, informing comparisons with LIGO/Virgo data.

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.