Covariant formulation of at finite speed propagating electric interaction of moving charges in Euclidean geometry

TL;DR

This paper develops a covariant, Galilean-invariant force law for moving charges in Euclidean geometry, revealing that Weber's law naturally emerges as an approximation of retarded electric interactions.

Contribution

It introduces a new retarded potential (CREP) and derives a Galilean-invariant Weber's force law, expanding the understanding of electromagnetic interactions beyond Maxwell's framework.

Findings

Weber's force law is recovered as a second-order approximation of CREP.

Magnetic forces are shown to be manifestations of retarded electric interactions.

Third-order approximation suggests the possibility of electromagnetic waves in Euclidean space.

Abstract

Maxwell's electrodynamics postulates the finite propagation speed of electromagnetic (EM) action and the notion of EM fields, but it only satisfies the requirement of the covariance in Minkowski metric (Lorentz invariance). Darwin's force law of moving charges, which originates from Maxwell's field theory complies the Lorentz invariance as well. Poincar\'e's principle stating that physical laws can be formulated with identical meaning on different geometries suggest, that the retarded EM interaction of moving charges might be covariant even in Euclidean geometry (Galilean invariance). Keeping the propagation speed finite, but breaking with Maxwell's field theory in this study an attempt is made to find a Galilean invariant force law. Through the altering of the Li\'enard-Wiechert potential (LWP) a new retarded potential of two moving charges, the Common Retarded Electric Potential (CREP) is introduced which depends on the velocities of both interacting charges. The sought after force law is determined by means of the second order approximation of CREP. The law obtained is the Galilean invariant Weber's force law, surprisingly. Its rediscovery from the second order approximation of a retarded electric potential confirms the significance of Weber's force law and proves it to be a retarded and approximative law. The fact that Weber's force law implies even the magnetic forces tells us that magnetic phenomena are a manifestation of the retarded electric interaction exclusively. The third order approximation of the CREP opens the possibility of EM waves, and the creation of a complete, Euclidean electrodynamics.