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→∞ (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.Comment: Updated a reference: Blanchet, Faye & Larrouturou 2022 in CQ