

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.

