Harmonic

Quantum mechanics

Curvature

Gravitation

Mathematical physics

Ricci curvature

Multipole expansion

Harmonic coordinates

Economics

Quadrupole

Classical mechanics

Physics

Fast multipole method

Mathematics

Astronomy

Quantum electrodynamics

Angular momentum

Mathematical analysis

Operations management

Gravitational wave

Geometry

Metric (unit)

Spherical multipole moments

Journal of High Energy Physics

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.

Multipole expansion of gravitational waves: from harmonic to Bondi coordinates