One must first assimilate the basic theory of bitensors (Part I), then apply the theory to construct convenient coordinate systems to chart a neighbourhood of the particle’s world line (Part II).

The reader is also assumed to have unlimited stamina, for the road to the equations of motion is a long one. The reader, however, is assumed to have a solid grasp of differential geometry and a deep understanding of general relativity. The presentation is entirely self-contained, and all relevant materials are developed ab initio. In this review we derive the equations that govern the motion of a point particle in a curved background spacetime. The self-force is proportional to q 2 in the case of a scalar charge, proportional to e 2 in the case of an electric charge, and proportional to m 2 in the case of a point mass. In curved spacetime the self-force contains a radiation-reaction component that is directly associated with dissipative effects, but it contains also a conservative component that is not associated with energy or angular-momentum transport. The particle’s motion is affected by the near-zone field which acts directly on the particle and produces a self-force. The radiation removes energy and angular momentum from the particle, which then undergoes a radiation reaction - its world line cannot be simply a geodesic of the background spacetime. These particles carry with them fields that behave as outgoing radiation in the wave zone. This article reviews the achievements described in the preceding paragraph it is concerned with the motion of a point scalar charge q, a point electric charge e, and a point mass m in a specified background spacetime with metric g αβ. The case of a point scalar charge was finally considered by Quinn in 2000, and this led to the realization that the mass of a scalar particle is not necessarily a constant of the motion.

In 1997 the motion of a point mass in a curved background spacetime was investigated by Mino, Sasaki, and Tanaka, who derived an expression for the particle’s acceleration (which is not zero unless the particle is a test mass) the same equations of motion were later obtained by Quinn and Wald using an axiomatic approach. (The field’s early history is well related in Ref.) In 1960 DeWitt and Brehme generalized Dirac’s result to curved spacetimes, and their calculation was corrected by Hobbs several years later. The motion of a point electric charge in flat spacetime was the subject of active investigation since the early work of Lorentz, Abrahams, Poincaré, and Dirac, until Gralla, Harte, and Wald produced a definitive derivation of the equations motion with all the rigour that one should demand, without recourse to postulates and renormalization procedures. 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. The review presents a detailed derivation of each of the three equations of motion (Part IV). It continues with a thorough discussion of Green’s functions in curved spacetime (Part III). It then applies the theory to the construction of convenient coordinate systems to chart a neighbourhood of the particle’s word line (Part II). The review begins with a discussion of the basic theory of bitensors (Part I). 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. 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. What remains after subtraction is a regular field that is fully responsible for the self-force. 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. 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. The work done by the self-force matches the energy radiated away by the particle. The self-force contains both conservative and dissipative terms, and the latter are responsible for the radiation reaction. 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. 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. 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.

Area of a surface in curved space