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

TL;DR

This paper presents a pre-metric approach to the Fresnel wave and ray surfaces, clarifying their geometric nature without relying on Euclidean metrics, and explains how electromagnetic waves induce spatial structures.

Contribution

It introduces a pre-metric formulation of Fresnel surfaces, elucidating the transition from tangent to cotangent spaces in electromagnetism without metric dependence.

Findings

Clarifies the geometric distinction between wave and ray surfaces.

Shows how electromagnetic waves induce spatial frames and metrics.

Provides a framework for pre-metric electromagnetism.

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.