Multipole expansion of gravitational waves: from harmonic to Bondi coordinates

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.

The fifth-order post-Newtonian Hamiltonian dynamics of two-body systems from an effective field theory approach: Potential contributions

“…(4.16'). 11 The associated D-dimensional terms up to the constant parts were calculated in [16,28], see also [37,47], and do also agree. This concerns the 1PN correction to the electric quadrupole moment, EQ ij Q ji , [37], the octupole moment, EO ijk O ijk , and the magnetic quadrupole moment, EJ ij J ji .…”

The Weyl BMS group and Einstein’s equations

“…However, it does not go as far as introducing a linear divergence in g uu and thus a time dependence of the leading order metric. One reason for this is that g uu = O(1) appears to be sufficient to describe the physics of compact binaries [91,92]. 5 The consequences of these different boundary conditions on the asymptotic symmetry group are discussed in section 4.…”

“…The Einstein's equations can be perturbatively solved in harmonic gauge under the assumption of no incoming radiation[91]. Transforming to Bondi or Newman-Unti coordinates one obtains guu = O(1)[92].…”

Conservation and integrability in lower-dimensional gravity

“…A widely used gauge to study asymptotic boundaries is the Bondi gauge [1,2,32,33]. Indeed, it is particularly well-adapted to investigate the interplay between radiation and symmetries [17,19,28,[34][35][36][37][38][39]. Furthermore, it allows us to consider simultaneously asymptotically locally flat spacetimes exhibiting null boundaries and asymptotically locally AdS spacetimes with timelike boundaries [26,[40][41][42][43][44][45].…”