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

TL;DR

The paper resolves two fundamental issues in the post-Newtonian treatment of isolated systems: divergent Poisson integrals and the near-zone limitation that obstructs enforcing no-incoming radiation. It develops a generalized Poisson operator with finite-part regularization and employs matched asymptotic expansions to connect the PN near-zone field with the exterior multipolar-post-Minkowskian field, enabling an algorithm to iterate the PN series to arbitrary order. Through exact matching, it determines the exterior multipole moments $X^{ ueta}_L(t)$ and the radiation-reaction functions $A^{ ueta}_L(t)$ as functionals of the source’s pseudo-tensor $ au^{ ueta}$, incorporating tail effects. The framework confirms the harmonic-coordinate condition as a consequence of stress-energy conservation and yields a coherent, self-consistent description of gravitational radiation and reaction for isolated systems in general relativity.

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.