Summary: 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 self-force contains both conservative and dissipative terms, and the latter are responsible for the radiation reaction. The work done by the self-force matches the energy radiated away by the particle. 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 – its only effect is to contribute to the particle’s inertia. What remains after subtraction is a regular field that is fully responsible for the self-force. Because this field satisfies a homogeneous wave equation, it can be thought of as a free field that interacts with the particle; it is this interaction that gives rise to 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 (Part I). It then applies the theory to the construction of convenient coordinate systems to chart a neighbourhood of the particle’s word line (Part II). It continues with a thorough discussion of Green’s functions in curved spacetime (Part III). The review presents a detailed derivation of each of the three equations of motion (Part IV). Because the notion of a point mass is problematic in general relativity, the review concludes (Part V) with an alternative derivation of the equations of motion that applies to a small body of arbitrary internal structure.
Update to the authors’ paper [Zbl 1071.83011]: This version of the review is a major update of the original article published in 2004. Two additional authors, Adam Pound and Ian Vega, have joined the article’s original author, and each one has contributed a major piece of the update. The literature survey presented in Sections 2 was contributed by Ian Vega, and Part V (Sections 20 to 23) was contributed by Adam Pound. Part V replaces a section of the 2004 article in which the motion of a small black hole was derived by the method of matched asymptotic expansions; this material can still be found in Ref. [142], but Pound’s work provides a much more satisfactory foundation for the gravitational self-force. The case study of Section 1.10 is new, and the “exact” formulation of the dynamics of a point mass in Section 19.1 is a major improvement from the original article. The concluding remarks of Section 24, contributed mostly by Adam Pound, are also updated from the 2004 article. The number of references has increased from 64 to 187.