### 1.1 Invitation

The motion of a point electric charge in flat spacetime was the subject of active investigation since the
early work of Lorentz, Abrahams, and Poincaré, until Dirac [25] produced a proper relativistic derivation
of the equations of motion in 1938. (The field’s early history is well related in [52].) In 1960 DeWitt
and Brehme [24] generalized Dirac’s result to curved spacetimes, and their calculation was
corrected by Hobbs [29] several years later. In 1997 the motion of a point mass in a curved
background spacetime was investigated by Mino, Sasaki, and Tanaka [39], 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 [49] using an axiomatic
approach. The case of a point scalar charge was finally considered by Quinn in 2000 [48], and this
led to the realization that the mass of a scalar particle is not necessarily a constant of the
motion.
This article reviews the achievements described in the preceding paragraph; it is concerned with
the motion of a point scalar charge , a point electric charge , and a point mass in
a specified background spacetime with metric . These particles carry with them fields
that behave as outgoing radiation in the wave zone. 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. The particle’s motion is affected by the
near-zone field which acts directly on the particle and produces a self-force. 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 self-force is proportional to in the case of a scalar charge,
proportional to in the case of an electric charge, and proportional to in the case of a point
mass.

In this review I derive the equations that govern the motion of a point particle in a curved background
spacetime. The presentation is entirely self-contained, and all relevant materials are developed ab initio. The
reader, however, is assumed to have a solid grasp of differential geometry and a deep understanding of
general relativity. The reader is also assumed to have unlimited stamina, for the road to the equations of
motion is a long one. One must first assimilate the basic theory of bitensors (Section 2), then apply the
theory to construct convenient coordinate systems to chart a neighbourhood of the particle’s world line
(Section 3). One must next formulate a theory of Green’s functions in curved spacetimes (Section 4), and
finally calculate the scalar, electromagnetic, and gravitational fields near the world line and figure out how
they should act on the particle (Section 5). The review is very long, but the payoff, I hope, will be
commensurate.

In this introductory section I set the stage and present an impressionistic survey of what the review
contains. This should help the reader get oriented and acquainted with some of the ideas and some of the
notation. Enjoy!