A pre-metric formulation of the Fresnel wave and ray surfaces

Abstract

The conventional formulation of the Fresnel wave and ray surfaces typically involves the implicit use of Euclidian metrics in order to justify treating vectors and covectors as indistinguishable. This tends to disguise the fact that one of the surfaces lives in the cotangent spaces to the spatial (or space-time) manifold, while the other lives in the tangent spaces. Moreover, no mention is made of how to get from tangent spaces to cotangent spaces in the absence of a metric. The following analysis shows how to resolve those issues within the framework of pre-metric electromagnetism. The way that electromagnetic waves imply spatial frames, coframes, and metrics will be explained.