The geometrical origin of the Doppler factor in the Liénard–Wiechert potentials

Abstract

We present an in depth analysis and a new derivation of the Doppler factor in the Liénard–Wiechert (LW) potentials, based on geometrical considerations in Minkowski space. We argue that, contrary to a common assumption, the methods used for deriving the Doppler factor in the case of an electrically charged extended particle are not applicable in the case of a point particle. A geometrical interpretation of the Doppler factor is nonetheless found in the later case, based on the electromagnetic interaction model of Fokker. In this model the interaction takes place between infinitesimal worldline segments with end points connected by light signals, just like the points where light pulses are emitted and detected in general. This analogy reveals that the relativistic Doppler effect is the missing link between the classical Doppler effect and the Doppler factor in the LW potentials.