Second-order gravitational self-force

TL;DR

This paper derives the second-order equation of motion for a small body in a vacuum spacetime using matched asymptotic expansions, revealing that the body's motion is geodesic in a specially defined regular geometry.

Contribution

It introduces a rigorous method to derive second-order gravitational self-force equations and outlines a numerical approach to compute the relevant metrics.

Findings

The body's motion is geodesic in a regular geometry satisfying Einstein's equations at second order.

A method is outlined for numerically obtaining the regular geometry and metric perturbations.

The approach neglects effects of the body's internal structure.

Abstract

Using a rigorous method of matched asymptotic expansions, I derive the equation of motion of a small, compact body in an external vacuum spacetime through second order in the body's mass (neglecting effects of internal structure). The motion is found to be geodesic in a certain locally defined regular geometry satisfying Einstein's equation at second order. I outline a method of numerically obtaining both the metric of that regular geometry and the complete second-order metric perturbation produced by the body.