The Quadrupole Moment of Compact Binaries to the Fourth post-Newtonian Order: Relating the Harmonic and Radiative Metrics

TL;DR

This paper compares harmonic and radiative metric constructions at 4PN order for compact binaries, establishing their equivalence and deriving relations between their parameters to improve gravitational-wave flux calculations.

Contribution

It introduces a method to relate harmonic and radiative metrics at 4PN order, facilitating more efficient tail-of-memory effect computations in gravitational-wave analysis.

Findings

Established the equivalence of harmonic and radiative metrics at 4PN order.

Derived the relation between canonical moments in both metric constructions.

Provided the 4PN order relation for the mass quadrupole moment.

Abstract

Motivated by the completion of the fourth post-Newtonian (4PN) gravitational-wave generation from compact binary systems, we analyze and contrast different constructions of the metric outside an isolated system, using post-Minkowskian expansions. The metric in "harmonic" coordinates has been investigated previously, in particular to compute tails and memory effects. However, it is plagued by powers of the logarithm of the radial distance $r$ when $r\to\infty$ (with $t-r/c=$ const). As a result, the tedious computation of the "tail-of-memory" effect, which enters the gravitational-wave flux at 4PN order, is more efficiently performed in the so-called "radiative" coordinates, which admit a (Bondi-type) expansion at infinity in simple powers of $r^{-1}$, without any logarithms. Here we consider a particular construction, performed order by order in the post-Minkowskian expansion, which directly yields a metric in radiative coordinates. We relate both constructions, and prove that they are physically equivalent as soon as a relation between the "canonical" moments which parametrize the radiative metric, and those parametrizing the harmonic metric, is verified. We provide the appropriate relation for the mass quadrupole moment at 4PN order, which will be crucial when deriving the "tail-of-memory" contribution to the gravitational flux.