Gravitational radiation reaction and second order perturbation theory

TL;DR

This paper develops a second-order perturbation theory framework for gravitational self-force, extending first-order methods to account for nonlinear effects in a small mass's motion through curved spacetime.

Contribution

It introduces a second-order perturbation approach, including a wave equation and regularization scheme, for analyzing gravitational self-force beyond linear approximation.

Findings

Derived a wave equation for second-order metric perturbations.

Established a regularization method for second-order self-force calculations.

Showed that the particle's motion is a geodesic in the perturbed metric including second-order effects.

Abstract

A point particle of small mass m moves in free fall through a background vacuum spacetime metric g_ab and creates a first-order metric perturbation h^1ret_ab that diverges at the particle. Elementary expressions are known for the singular m/r part of h^1ret_ab and for its tidal distortion determined by the Riemann tensor in a neighborhood of m. Subtracting this singular part h^1S_ab from h^1ret_ab leaves a regular remainder h^1R_ab. The self-force on the particle from its own gravitational field adjusts the world line at O(m) to be a geodesic of g_ab+h^1R_ab. The generalization of this description to second-order perturbations is developed and results in a wave equation governing the second-order h^2ret_ab with a source that has an O(m^2) contribution from the stress-energy tensor of m added to a term quadratic in h^1ret_ab. Second-order self-force analysis is similar to that at first order: The second-order singular field h^2S_ab subtracted from h^2ret_ab yields the regular remainder h^2R_ab, and the second-order self-force is then revealed as geodesic motion of m in the metric g_ab+h^1R+h^2R.