Symmetries and pre-metric electromagnetism

TL;DR

This paper explores the symmetries of pre-metric electromagnetism formulated as an exterior differential system, analyzing how different constitutive laws affect the symmetry Lie algebra and its possible extensions.

Contribution

It provides a detailed computation of the symmetry equations for various electromagnetic constitutive laws within the pre-metric framework, revealing how these laws influence symmetry structures.

Findings

Four prolongation methods of the symmetry Lie algebra in the uniform linear case.

Deformation of symmetry Lie algebra in the uniform nonlinear case due to field strengths.

Inconclusive effects of non-uniformity on symmetry without further specifics.

Abstract

The equations of pre-metric electromagnetism are formulated as an exterior differential system on the bundle of exterior differential 2-forms over the spacetime manifold. The general form for the symmetry equations of the system is computed and then specialized to various possible forms for an electromagnetic constitutive law, namely, uniform linear, non-uniform linear, and uniform nonlinear. It is shown that in the uniform linear case, one has four possible ways of prolonging the symmetry Lie algebra, including prolongation to a Lie algebra of infinitesimal projective transformations of a real four-dimensional projective space. In the most general non-uniform linear case, th effect of non-uniformity on symmetry seems inconclusive in the absence of further specifics, and in the uniform nonlinear case, the overall difference from the uniform linear case amounts to a deformation of the electromagnetic constitutive tensor by the electromagnetic fields strengths, which induces a corresponding deformation of the symmetry Lie algebra that was obtained in the uniform linear case.