The algebraic origin of the Doppler factor in the Lienard-Wiechert potentials

TL;DR

This paper explores the algebraic derivation of the Doppler factor in Lienard-Wiechert potentials, emphasizing the importance of the weak Dirac delta function and the segment-based nature of electromagnetic interactions in spacetime.

Contribution

It clarifies the algebraic origin of the Doppler factor and the role of the weak Dirac delta in electrodynamics, linking it to the segment-based interaction along worldlines.

Findings

The Doppler factor arises from the weak Dirac delta function.

Electromagnetic interactions occur between infinitesimal segments, not points.

The derivation relies on the superposition principle and action formulations.

Abstract

After reviewing the algebraic derivation of the Doppler factor in the Lienard-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.