# Nonlinear gravitational self-force: second-order equation of motion

## TL;DR

This paper derives the second-order gravitational self-force equation of motion for a small compact object in a background spacetime, extending previous results to more regular gauges suitable for numerical calculations.

## Contribution

It provides a detailed derivation of the second-order self-force equation and extends the gauge class for improved numerical self-force computations.

## Key findings

- Second-order motion follows a geodesic in a smooth vacuum metric.
- Extension to highly regular gauges for numerical applications.
- Complete derivation of second-order self-force equation.

## Abstract

When a small, uncharged, compact object is immersed in an external background spacetime, at zeroth order in its mass it moves as a test particle in the background. At linear order, its own gravitational field alters the geometry around it, and it moves instead as a test particle in a certain effective metric satisfying the linearized vacuum Einstein equation. In the letter [Phys. Rev. Lett. 109, 051101 (2012)], using a method of matched asymptotic expansions, I showed that the same statement holds true at second order: if the object's leading-order spin and quadrupole moment vanish, then through second order in its mass it moves on a geodesic of a certain smooth, locally causal vacuum metric defined in its local neighbourhood. Here I present the complete details of the derivation of that result. In addition, I extend the result, which had previously been derived in gauges smoothly related to Lorenz, to a class of highly regular gauges that should be optimal for numerical self-force computations.