On the derivation of the spacetime metric from linear electrodynamics

In the framework of metric-free electrodynamics, we start with a {\em linear} spacetime relation between the excitation 2-form $H=(\mathcal{D},\mathcal{H})$ 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.