Second gradient electrodynamics: a non-singular relativistic field theory

TL;DR

This paper develops a covariant, non-singular relativistic field theory called second gradient electrodynamics, extending classical electrodynamics with higher-order derivatives, providing explicit solutions for point charges and analyzing wave propagation.

Contribution

It introduces a second gradient electrodynamics framework with explicit non-singular solutions and a relativistic equation of motion for charged particles, extending classical theory with higher-order derivatives.

Findings

Explicit non-singular potentials for point charges

Derived relativistic equations of motion with nonlocal self-force

Identified three wave modes, including non-dispersive and dispersive waves

Abstract

In this paper, we give the covariant formulation of second gradient electrodynamics, which is a generalized electrodynamics of second order including derivatives of higher order. The relativistic form of the field equations, the energy-momentum tensor and the Lorentz force density are presented. For an electric point charge, the generalized Lienard-Wiechert potentials and the corresponding electromagnetic field strength tensor are given as retarded integral expressions. Explicit formulas for the electromagnetic potential vector and electromagnetic field strength tensor of a uniformly moving point charge are found without any singularity and discontinuity. In addition, a world-line integral expression for the self-force of a charged point particle is given. The relativistic equation of motion of a charged particle coupled with electromagnetic fields in second gradient electrodynamics is derived, which is an integro-differential equation with nonlocality in time. For a uniformly accelerated charge, explicit formulas of the self-force and the electromagnetic mass, being non-singular, are given. Moreover, the wave propagation and the dispersion relations in the vacuum of second gradient electrodynamics are analyzed. Three modes of waves were found: one non-dispersive wave as in Maxwell electrodynamics, and two dispersive waves similar to the wave propagation in a collisionless plasma.