Determination of an electromagnetic medium from the Fresnel surface

We study Maxwell's equations on a 4-manifold where the electromagnetic medium is described by a suitable antisymmetric ((2)(2))-tensor kappa with real components. 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 generalization of the null cone determined by a Lorentz metric; the Fresnel surface parameterizes electromagnetic wavespeed as a function of direction. We show that if (a) kappa has no skewon and no axion component, (b) kappa is invertible and (c) the Fresnel surface is pointwise a Lorentz null cone, then the tensor kappa is proportional to a Hodge star operator of a Lorentz metric and kappa represents an isotropic medium. In other words, in a suitable class of media one can recognize isotropic media from wavespeed alone. What is more, we study the nonunique dependence between the tensor kappa, its Tamm-Rubilar tensor density and its Fresnel surface. For example, we show that if kappa is invertible, then kappa and kappa(-1) have the same Fresnel surfaces.