Determination of electromagnetic medium from the Fresnel surface

We study Maxwell's equations on a 4-manifold where the electromagnetic medium is described by an antisymmetric $2\choose 2$-tensor $κ$. 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 $κ$ has only a principal part and if the Fresnel surface of $κ$ coincides with the light cone for a Lorentz metric $g$, then $κ$ is proportional to the Hodge star operator of $g$. That is, under additional assumptions, the Fresnel surface of $κ$ determines the conformal class of $κ$. The purpose of this paper is twofold. First, we provide a new proof of this result using Gröbner 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 $κ$. For example, if $κ$ is invertible we show that $κ$ and $κ^{-1}$ have the same Fresnel surfaces.

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