Backwards on Minkowski's road. From 4D to 3D Maxwellian electromagnetism

TL;DR

This paper explores the fundamental structure of electromagnetism in Minkowski space, deriving four possible sets of equations for different media and analyzing their physical implications and invariants.

Contribution

It introduces four distinct covariant formulations of Maxwell's equations for isotropic media, revealing how sign assumptions and invariants vary across these formulations.

Findings

Four possible electromagnetic media types with distinct sign conventions.

Wave propagation described by Minkowskian optical metrics in all media.

Electric and magnetic energies are nonnegative in all cases.

Abstract

Minkowski's concept of a four-dimensional physical space is a central paradigm of modern physics. The three-dimensional Maxwellian electrodynamics is uniquely generalized to the covariant four-dimensional form. Is the (1+3) decomposition of the covariant four-dimensional form unique? How do the different sign assumptions of electrodynamics emerge from this decomposition? Which of these assumptions are fundamental and which of them may be modified? How does the Minkowski space-time metric emerge from this preliminary metric-free construction? In this paper we are looking for answers to the problems mentioned. Our main result is the derivation of four different possible sets of electrodynamic equations which may occur in different types of isotropic electromagnetic media. The wave propagation in each of these media is described by the Minkowskian optical metrics. Moreover, the electric and magnetic energies are nonnegative in all cases. We also show that the correct directions of the Lorentz force (as a consequence of the Dufay and the Lenz rules) hold true for all these cases. However, the differences between these four types of media must have a physical meaning. In particular, the signs of the three electromagnetic invariants are different.