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

Abstract

After reviewing the algebraic derivation of the Doppler factor in the Liénard–Wiechert potentials of an electrically charged point particle, we conclude that the Dirac delta function used in electrodynamics must be the one obeying the weak definition, non-zero in an infinitesimal neighborhood, and not the one obeying the strong definition, non-zero in a point. This conclusion emerges from our analysis of (a) the derivation of an important Dirac delta function identity, which generates the Doppler factor, (b) the linear superposition principle implicitly used by the Green function method, and (c) the two equivalent formulations of the Schwarzschild–Tetrode–Fokker action. As a consequence, in full agreement with our previous discussion of the geometrical origin of the Doppler factor, we conclude that the electromagnetic interaction takes place not between points in Minkowski space, but between corresponding infinitesimal segments along the worldlines of the particles.