Generally covariant Fresnel equation and the emergence of the light cone structure in linear pre-metric electrodynamics

Abstract

We study the {\em propagation of electromagnetic waves} in a spacetime devoid of a metric but equipped with a {\em linear} electromagnetic spacetime relation $H\simχ\cdot F$. Here $H$ is the electromagnetic excitation $({\cal D},{\cal H})$ and $F$ the field strength $(E,B)$, whereas $χ$ (36 independent components) characterizes the electromagnetic permittivity/permeability of spacetime. We derive analytically the corresponding Fresnel equation and show that it is always quartic in the wave covectors. We study the `Fresnel tensor density' ${\cal G}^{ijkl}$ as (cubic) function of $χ$ and identify the leading part of $χ$ (20 components) as indispensable for light propagation. Upon requiring electric/magnetic reciprocity of the spacetime relation, the leading part of $χ$ induces the {\em light cone} structure of spacetime (9 components), i.e., the spacetime metric up to a function. The possible existence of an Abelian {\em axion} field (1 component of $χ$) and/or of a {\em skewon} field (15 components) and their effect on light propagation is discussed in some detail. The newly introduced skewon field is expected to be T-odd and related to dissipation.