Multipole Expansion of Gravitational Waves: from Harmonic to Bondi coordinates

TL;DR

The paper addresses how to relate the multipolar post-Minkowskian metric expressed in harmonic (de Donder) coordinates to radiative Newman-Unti (NU) coordinates, producing an explicit diffeomorphism that maps the exterior metric to NU form order by order in the PM expansion. The authors solve the NU gauge conditions perturbatively, recover the generalized BMS symmetry under appropriate boundary conditions, and show that at linear order the Bondi data (mass and angular momentum aspects, Bondi shear) can be read off from the canonical multipoles. At quadratic order, they derive the NU metric for the mass-quadrupole interaction, including tail terms via the radiative quadrupole moment and demonstrate that the NU expansion is regular at null infinity while tails modify the radiative content. They also obtain the GW mass and angular-momentum losses in the Bondi-NU framework and discuss the connection to the standard quadrupole formula, setting the stage for generalization to higher multipole couplings and memory effects.

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.