On the derivation of the spacetime metric from linear electrodynamics

TL;DR

This paper derives the spacetime metric from linear, metric-free electrodynamics by analyzing the Fresnel equation, showing it reduces to a light cone, thus linking electromagnetic wave propagation to spacetime geometry.

Contribution

It extends previous work by solving the linear electrodynamics system including singular solutions and deriving the metric from wave propagation characteristics.

Findings

The Fresnel surface reduces to a light cone for all solutions.

The spacetime metric is derived up to a conformal factor.

Electromagnetic wave propagation determines the spacetime geometry.

Abstract

In the framework of metric-free electrodynamics, we start with a {\em linear} spacetime relation between the excitation 2-form $H = ({\cal D}, {\cal H})$ and the field strength 2-form $F = ({E,B})$. This linear relation is constrained by the so-called closure relation. We solve this system algebraically and extend a previous analysis such as to include also singular solutions. Using the recently derived fourth order {\em Fresnel} equation describing the propagation of electromagnetic waves in a general {\em linear} medium, we find that for all solutions the fourth order surface reduces to a light cone. Therefrom we derive the corresponding metric up to a conformal factor.