Determination of an electromagnetic medium from the Fresnel surface

Abstract

We study Maxwell’s equations on a 4-manifold where the electromagnetic medium is described by a suitable antisymmetric -tensor κ 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) κ has no skewon and no axion component, (b) κ is invertible and (c) the Fresnel surface is pointwise a Lorentz null cone, then the tensor κ is proportional to a Hodge star operator of a Lorentz metric and κ 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 κ, its Tamm–Rubilar tensor density and its Fresnel surface. For example, we show that if κ is invertible, then κ and κ−1 have the same Fresnel surfaces.