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

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.