Determination of electromagnetic medium from the Fresnel surface

TL;DR

This paper investigates how the Fresnel surface, derived from Maxwell's equations on a 4-manifold, can determine the electromagnetic medium tensor, providing new proofs and exploring cases where it does not uniquely identify the medium.

Contribution

It offers a new proof using Gr"obner bases that the Fresnel surface determines the conformal class of the medium tensor under certain conditions and identifies cases where this determination fails.

Findings

Fresnel surface coincides with light cone iff medium tensor is proportional to Hodge star

Inversion of the medium tensor preserves the Fresnel surface

Fresnel surface does not always uniquely determine the medium tensor

Abstract

We study Maxwell's equations on a 4-manifold where the electromagnetic medium is described by an antisymmetric $2\choose 2$-tensor $\kappa$. In this setting, the Tamm-Rubilar tensor density determines a polynomial surface of fourth order in each cotangent space. This surface is called the Fresnel surface and acts as a generalisation of the light-cone determined by a Lorentz metric; the Fresnel surface parameterises electromagnetic wave-speed as a function of direction. Favaro and Bergamin have recently proven that if $\kappa$ has only a principal part and if the Fresnel surface of $\kappa$ coincides with the light cone for a Lorentz metric $g$, then $\kappa$ is proportional to the Hodge star operator of $g$. That is, under additional assumptions, the Fresnel surface of $\kappa$ determines the conformal class of $\kappa$. The purpose of this paper is twofold. First, we provide a new proof of this result using Gr\"obner bases. Second, we describe a number of cases where the Fresnel surface does not determine the conformal class of the original $2\choose 2$-tensor $\kappa$. For example, if $\kappa$ is invertible we show that $\kappa$ and $\kappa^{-1}$ have the same Fresnel surfaces.