Abstract
The authors prove that if the source of a linearised Robinson-Trautman field (1962) is a simple pole particle in a background Minkowskian space-time, and if the motion of the particle is rectilinear in an inertial frame of reference then (i) if the particle is uncharged it must move with uniform acceleration, or (ii) if the particle is charged it must move either with uniform acceleration or perform run-away motion from a state of uniform acceleration in the infinite past.