Post-Newtonian approximation for isolated systems calculated by matched asymptotic expansions

TL;DR

This paper develops a new method to improve the post-Newtonian approximation in general relativity by resolving divergence issues and extending its validity through matched asymptotic expansions, enabling precise modeling of gravitational radiation.

Contribution

It introduces a divergence-free iterative scheme and a matching technique to extend the post-Newtonian approximation's applicability to include boundary conditions at infinity.

Findings

Resolved divergence issues with a new Poisson-type integral operator.

Extended the domain of validity of the post-Newtonian approximation.

Identified terms related to gravitational radiation reaction in the series.

Abstract

Two long-standing problems with the post-Newtonian approximation for isolated slowly-moving systems in general relativity are: (i) the appearance at high post-Newtonian orders of divergent Poisson integrals, casting a doubt on the soundness of the post-Newtonian series; (ii) the domain of validity of the approximation which is limited to the near-zone of the source, and prevents one, a priori, from incorporating the condition of no-incoming radiation, to be imposed at past null infinity. In this article, we resolve the problem (i) by iterating the post-Newtonian hierarchy of equations by means of a new (Poisson-type) integral operator that is free of divergencies, and the problem (ii) by matching the post-Newtonian near-zone field to the exterior field of the source, known from previous work as a multipolar-post-Minkowskian expansion satisfying the relevant boundary conditions at infinity. As a result, we obtain an algorithm for iterating the post-Newtonian series up to any order, and we determine the terms, present in the post-Newtonian field, that are associated with the gravitational-radiation reaction onto an isolated slowly-moving matter system.