Singular perturbation techniques in the gravitational self-force problem

TL;DR

This paper clarifies the mathematical foundations of singular perturbation techniques in general relativity, especially for gravitational self-force calculations, and evaluates their effectiveness compared to alternative methods.

Contribution

It explicates the geometrical structure of singular perturbation theory in relativity and refines the matching conditions used in self-force computations.

Findings

Matched asymptotic expansions often rely on weak matching conditions

Weak conditions may not uniquely determine equations of motion

Alternative methods yield stronger results for self-force problems

Abstract

Much of the progress in the gravitational self-force problem has involved the use of singular perturbation techniques. Yet the formalism underlying these techniques is not widely known. I remedy this situation by explicating the foundations and geometrical structure of singular perturbation theory in general relativity. Within that context, I sketch precise formulations of the methods used in the self-force problem: dual expansions (including matched asymptotic expansions), for which I identify precise matching conditions, one of which is a weak condition arising only when multiple coordinate systems are used; multiscale expansions, for which I provide a covariant formulation; and a self-consistent expansion with a fixed worldline, for which I provide a precise statement of the exact problem and its approximation. I then present a detailed analysis of matched asymptotic expansions as they have been utilized in calculating the self-force. Typically, the method has relied on a weak matching condition, which I show cannot determine a unique equation of motion. I formulate a refined condition that is sufficient to determine such an equation. However, I conclude that the method yields significantly weaker results than do alternative methods.