  We consider the classic problem of a compact fluid source that behaves
non-relativistically and that radiates gravitational waves. The problem
consists of determining the metric close to the source as well as far away from
it. The non-relativistic nature of the source leads to a separation of scales
resulting in an overlap region where both the $1/c$ and (multipolar)
$G$-expansions are valid. Standard approaches to this problem (the
Blanchet--Damour and the DIRE approach) use the harmonic gauge. We define a
`post-Newtonian' class of gauges that admit a Newtonian regime in inertial
coordinates. In this paper we set up a formalism to solve for the metric for
any post-Newtonian gauge choice. Our methods are based on previous work on the
covariant theory of non-relativistic gravity (a $1/c$-expansion of general
relativity that uses post-Newton-Cartan variables). At the order of interest in
the $1/c$ and $G$-expansions we split the variables into two sets: transverse
and longitudinal. We show that for the transverse variables the problem can be
reduced to inverting Laplacian and d'Alembertian operators on their respective
domains subject to appropriate boundary conditions. The latter are regularity
in the interior and asymptotic flatness with a Sommerfeld no-incoming radiation
condition imposed at past null infinity. The longitudinal variables follow from
the gauge choice. The full solution is then obtained by the method of matched
asymptotic expansion. We show that our methods reproduce existing results in
harmonic gauge to 2.5PN order.
