Asymptotic expansions for relativistic celestial mechanics

TL;DR

This paper develops asymptotic expansion methods for relativistic celestial mechanics, comparing post-Newtonian and post-Minkowskian approximations within a scalar gravity theory, highlighting their differences and limitations.

Contribution

It introduces a systematic asymptotic expansion scheme for relativistic celestial mechanics and compares two hypotheses for time variables, clarifying their implications and compatibility.

Findings

Asymptotic expansions can be applied to scalar relativistic gravity theories.

The standard post-Newtonian approximation is incompatible with the asymptotic expansion method used.

Post-Minkowskian approximation allows describing propagation effects but does not match the Newtonian limit.

Abstract

The method of asymptotic expansions is used to build an approximation scheme relevant to celestial mechanics in relativistic theories of gravitation. A scalar theory is considered, both as a simple example and for its own sake. This theory is summarized, then the relevant boundary problem is seen to be the full initial-value problem. It is shown that, with any given system of gravitating bodies, one may associate a one-parameter family of similar systems, the parameter measuring the gravitational field-strength. After a specific change of units, the derivation of asymptotic expansions becomes straightforward. Two hypotheses could be made as to which time variable has to be used in the expansion. The first one leads to an "asymptotic" post-Newtonian approximation (PNA) with instantaneous propagation, differing from the standard PNA in that, in the asymptotic PNA, all fields are expanded. The second hypothese could lead to an "asymptotic" post-Minkowskian approximation (PMA) allowing to describe propagation effects, but it is not compatible with the Newtonian limit. It is shown that the standard PNA is not compatible with the application of the usual method of asymptotic expansions as envisaged here.