Non-dissipative electromagnetic media with two Lorentz null cones

We study Maxwell's equations on a 4-manifold where the electromagnetic medium is modeled by an antisymmetric 22-tensor with 21 real coefficients. In this setting the Fresnel surface is a fourth-order polynomial surface that describes the dynamical response of the medium in the geometric optics limit. For example, in an isotropic medium the Fresnel surface is a Lorentz null cone. The contribution of this paper is the pointwise description of all electromagnetic medium tensors κ with real coefficients that satisfy the following three conditions:

medium κ is invertible,

medium κ is skewon-free, or non-dissipative,

the Fresnel surface of κ is the union of two distinct Lorentz null cones.

We show that there are only three classes of media with these properties and give explicit expressions in local coordinates for each class.