Maxwell's equations in Minkowski's world: their premetric generalization and the electromagnetic energy-momentum tensor

TL;DR

This paper explores the historical development and modern generalization of Minkowski's formulation of Maxwell's equations, emphasizing their premetric form and the electromagnetic energy-momentum tensor within tensor calculus and general relativity.

Contribution

It provides a detailed analysis of Minkowski's original derivation and extends Maxwell's equations to a premetric framework applicable in general relativity.

Findings

Generalization of Maxwell's equations to premetric form

Derivation of the electromagnetic energy-momentum tensor in premetric form

Applications of premetric electrodynamics in modern physics

Abstract

In December 1907, Minkowski expressed the Maxwell equations in the very beautiful and compact 4-dimensional form: lor f=-s, lor F^*=0. Here `lor', an abbreviation of Lorentz, represents the 4-dimensional differential operator. We study Minkowski's derivation and show how these equations generalize to their modern premetric form in the framework of tensor and exterior calculus (valid also in general relativity). After mentioning some applications of premetric electrodynamics, we turn to Minkowski's discovery of the energy-momentum tensor of the electromagnetic field. We discuss how he arrived at it and how its premetric formulation looks like.