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We consider the classic problem of a compact fluid source that behavesnon-relativistically and that radiates gravitational waves. The problemconsists of determining the metric close to the source as well as far away fromit. The non-relativistic nature of the source leads to a separation of scalesresulting in an overlap region where both the $1/c$ and (multipolar)$G$-expansions are valid. Standard approaches to this problem (theBlanchet--Damour and the DIRE approach) use the harmonic gauge. We define a`post-Newtonian' class of gauges that admit a Newtonian regime in inertialcoordinates. In this paper we set up a formalism to solve for the metric forany post-Newtonian gauge choice. Our methods are based on previous work on thecovariant theory of non-relativistic gravity (a $1/c$-expansion of generalrelativity that uses post-Newton-Cartan variables). At the order of interest inthe $1/c$ and $G$-expansions we split the variables into two sets: transverseand longitudinal. We show that for the transverse variables the problem can bereduced to inverting Laplacian and d'Alembertian operators on their respectivedomains subject to appropriate boundary conditions. The latter are regularityin the interior and asymptotic flatness with a Sommerfeld no-incoming radiationcondition imposed at past null infinity. The longitudinal variables follow fromthe gauge choice. The full solution is then obtained by the method of matchedasymptotic expansion. We show that our methods reproduce existing results inharmonic gauge to 2.5PN order.

Towards a covariant framework for post-Newtonian expansions for radiative sources