### 1.8 Motion of a scalar charge in curved spacetime

The dynamics of a point scalar charge can be formulated in a way that stays fairly close to the
electromagnetic theory. The particle’s charge produces a scalar field , which satisfies a wave
equation
that is very similar to Equation (13). Here, is the spacetime’s Ricci scalar, and is an
arbitrary coupling constant; the scalar charge density is given by a four-dimensional Dirac
functional supported on the particle’s world line . The retarded solution to the wave equation is
where is the retarded Green’s function associated with Equation (34). The field exerts a force on
the particle, whose equations of motion are
where is the particle’s mass; this equation is very similar to the Lorentz-force law. But the dynamics of
a scalar charge comes with a twist: If Equations (34) and (36) are to follow from a variational principle, the
particle’s mass should not be expected to be a constant of the motion. It is found instead to satisfy the
differential equation
and in general will vary with proper time. This phenomenon is linked to the fact that a scalar field has
zero spin: The particle can radiate monopole waves and the radiated energy can come at the expense of the
rest mass.
The scalar field of Equation (35) diverges on the world line, and its singular part must be
removed before Equations (36) and (37) can be evaluated. This procedure produces the radiative field
, and it is this field (which satisfies the homogeneous wave equation) that gives rise to a self-force.
The gradient of the radiative field takes the form of

when it is evaluated of the world line. The last term is the tail integral
and this brings the dependence on the particle’s past.
Substitution of Equation (38) into Equations (36) and (37) gives the equations of motion of a point
scalar charge. (At this stage I introduce an external force and reduce the order of the differential
equation.) The acceleration is given by

and the mass changes according to
These equations were first derived by
Quinn [48].
In flat spacetime the Ricci-tensor term and the tail integral disappear, and Equation (40) takes the
form of Equation (5) with replacing the factor of . In this simple case
Equation (41) reduces to and the mass is in fact a constant. This property remains true in
a conformally-flat spacetime when the wave equation is conformally invariant ():
In this case the Green’s function possesses only a light-cone part, and the right-hand side of
Equation (41) vanishes. In generic situations the mass of a point scalar charge will vary with proper
time.