Radiation reaction and the self-force for a point mass in general relativity

TL;DR

This paper investigates the self-force experienced by a point mass in general relativity, deriving expressions for the metric perturbation and its regularized form, and providing an approximate solution for small black holes with minimal error.

Contribution

It introduces explicit formulas for the singular and regular parts of the metric perturbation caused by a point mass, advancing understanding of gravitational self-force calculations.

Findings

Derived simple expressions for the singular metric perturbation.

Defined a regularized metric perturbation that is C^1.

Provided an approximate Einstein solution with O(m^2) accuracy for small black holes.

Abstract

A point particle of mass m moving on a geodesic creates a perturbation h, of the spacetime metric g, that diverges at the particle. Simple expressions are given for the singular m/r part of h and its quadrupole distortion caused by the spacetime. Subtracting these from h leaves a remainder h^R that is C^1. The self-force on the particle from its own gravitational field corrects the worldline at O(m) to be a geodesic of g+h^R. For the case that the particle is a small non-rotating black hole, an approximate solution to the Einstein equations is given with error of O(m^2) as m approaches 0.