Multipole Expansion of Gravitational Waves: from Harmonic to Bondi coordinates

TL;DR

This paper develops a detailed transformation of gravitational wave metrics from harmonic to Newman-Unti coordinates, analyzing multipole moments, residual symmetries, and tail effects to improve understanding of gravitational radiation in different coordinate systems.

Contribution

It provides a new explicit transformation of the gravitational wave metric in multipolar post-Minkowskian approximation from harmonic to NU coordinates, including tail effects and symmetry analysis.

Findings

Derived NU metric as a functional of multipole moments

Identified residual symmetry group in NU coordinates

Analyzed tail contributions to the quadrupole interaction

Abstract

We transform the metric of an isolated matter source in the multipolar post-Minkowskian approximation from harmonic (de Donder) coordinates to radiative Newman-Unti (NU) coordinates. To linearized order, we obtain the NU metric as a functional of the mass and current multipole moments of the source, valid all-over the exterior region of the source. Imposing appropriate boundary conditions we recover the generalized Bondi-van der Burg-Metzner-Sachs residual symmetry group. To quadratic order, in the case of the mass-quadrupole interaction, we determine the contributions of gravitational-wave tails in the NU metric, and prove that the expansion of the metric in terms of the radius is regular to all orders. The mass and angular momentum aspects, as well as the Bondi shear, are read off from the metric. They are given by the radiative quadrupole moment including the tail terms.