Linear pre-metric electrodynamics and deduction of the light cone

TL;DR

This paper develops a covariant framework for linear pre-metric electrodynamics, deriving conditions under which a light cone structure and conformal metric emerge from the constitutive tensor of spacetime.

Contribution

It provides a general derivation of the Fresnel equation for arbitrary linear constitutive tensors and explores conditions for the emergence of a light cone and conformal metric from these tensors.

Findings

Closure and symmetry of the constitutive tensor imply the existence of a conformal metric.

Explicit metric components can be deduced from the constitutive tensor under certain conditions.

Relaxing symmetry conditions affects the emergence of the light cone structure.

Abstract

We formulate a general framework for describing the electromagnetic properties of spacetime. These properties are encoded in the `constitutive tensor of the vacuum', a quantity analogous to that used in the description of material media. We give a generally covariant derivation of the Fresnel equation describing the local properties of the propagation of electromagnetic waves for the case of the most general possible linear constitutive tensor. We also study the particular case in which a light cone structure is induced and the circumstances under which such a structure emerges. In particular, we will study the relationship between the dual operators defined by the constitutive tensor under certain conditions and the existence of a conformal metric. Closure and symmetry of the constitutive tensor will be found as conditions which ensure the existence of a conformal metric. We will also see how the metric components can be explicitly deduced from the constitutive tensor if these two conditions are met. Finally, we will apply the same method to explore the consequences of relaxing the condition of symmetry and how this affects the emergence of the light cone.