Constructing the self-force

TL;DR

This paper reviews methods for computing the scalar self-force on a point particle in curved spacetime, focusing on geometric tools, coordinate systems, wave equations, and regularization techniques, with insights into improving the point particle model.

Contribution

It provides a comprehensive overview of the mathematical and conceptual framework for calculating the scalar self-force, including regularization and axiomatic foundations.

Findings

Describes the use of Green's functions to solve wave equations in curved spacetime.

Explains how to regularize the self-field to obtain a finite self-force.

Discusses potential improvements beyond the point particle approximation.

Abstract

I present an overview of the methods involved in the computation of the scalar, electromagnetic, and gravitational self-forces acting on a point particle moving in a curved spacetime. For simplicity, the focus here will be on the scalar self-force. The lecture follows closely my review article on this subject published in Living Reviews in Relativity. I begin with a review of geometrical elements (Synge's world function, the parallel propagator). Next I introduce useful coordinate systems (Fermi normal coordinates and retarded light-cone coordinates) in a neighborhood of the particle's world line. I then present the wave equation for a scalar field in curved spacetime and the equations of motion for a particle endowed with a scalar charge. The wave equation is solved by means of a Green's function, and the self-force is constructed from the field gradient. Because the retarded field is singular on the world line, the self-force must involve a regularized version of the field gradient, and I describe how the regular piece of the self-field can be identified. In the penultimate section of the lecture I put the construction of the self-force on a sophisticated axiomatic basis, and in the concluding section I explain how one can do better by abandoning the dangerous fiction of a point particle