The motion of point particles in curved spacetime

TL;DR

This review comprehensively discusses the derivation of equations of motion for point particles with scalar, electric, and mass in curved spacetime, emphasizing self-force effects and mathematical tools like bitensors and Green's functions.

Contribution

It develops from scratch the mathematical framework for deriving particle equations of motion in curved spacetime, including self-force calculations and handling singular fields.

Findings

Isolated the singular part of the field exerts no force

Derived detailed equations of motion for scalar, electric, and mass particles

Presented an alternative approach for small bodies with internal structure

Abstract

This review is concerned with the motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime. In each of the three cases the particle produces a field that behaves as outgoing radiation in the wave zone, and therefore removes energy from the particle. In the near zone the field acts on the particle and gives rise to a self-force that prevents the particle from moving on a geodesic of the background spacetime. The field's action on the particle is difficult to calculate because of its singular nature: the field diverges at the position of the particle. But it is possible to isolate the field's singular part and show that it exerts no force on the particle. What remains after subtraction is a smooth field that is fully responsible for the self-force. The mathematical tools required to derive the equations of motion of a point scalar charge, a point electric charge, and a point mass in a specified background spacetime are developed here from scratch. The review begins with a discussion of the basic theory of bitensors. It then applies the theory to the construction of convenient coordinate systems to chart a neighbourhood of the particle's word line. It continues with a thorough discussion of Green's functions in curved spacetime. The review presents a detailed derivation of each of the three equations of motion. Because the notion of a point mass is problematic in general relativity, the review concludes with an alternative derivation of the equations of motion that applies to a small body of arbitrary internal structure.