Post-Newtonian approximation for isolated systems by matched asymptotic expansions I. General structure revisited

TL;DR

This paper revisits and corrects previous work on post-Newtonian approximations in general relativity, demonstrating that exact matching of asymptotic expansions is inevitable and providing a revised framework for such analysis.

Contribution

It identifies and corrects an error in prior matching procedures, establishing that the asymptotic match must be exact and offering a new approach to the problem.

Findings

Corrected the matching process in post-Newtonian approximations

Proved that the asymptotic match is necessarily exact

Validated the corrected method with a model problem

Abstract

In recent years post-Newtonian approximations for isolated slowly-moving systems in general relativity have been studied by means of matched asymptotic expansions. A paper by Poujade & Blanchet in 2002 made great progress by effectively reducing the use of such expansions to an algorithmic form. It gave systematic procedures for the development of both near-zone and far-zone asymptotic expansions, avoiding the divergent integrals which often bedevilled such methods, and showed that these two expansions could be made to match exactly, a result described there as somewhat remarkable. This paper revisits that work and shows that there is unfortunately an error in it which invalidates the results of the matching process as given therein. The present paper identifies that error and shows how it may be corrected to give valid matching results. The correction is presented in a redevelopment somewhat different from that of their paper. This shows that far from being somewhat remarkable, it is in fact inevitable that the match is exact. It is indeed remarkable that they could carry both expansions to the point at which matching becomes possible, but if it can be done at all, then the match is necessarily exact. A companion paper will apply this asymptotic matching to a model problem in which the correct near-zone expansion was obtained by approximation from an exact solution. It will be shown that there is a discrepancy between this expansion and results from the original development of Poujade & Blanchet but that the corrected development presented here reproduces the result of the model problem exactly.