Spacetime metric from local and linear electrodynamics: a new axiomatic   scheme
  We consider spacetime to be a 4-dimensional differentiable manifold that can be split locally into time and space. No metric, no linear connection are assumed. Matter is described by classical fields/fluids. We distinguish electrically charged from neutral matter. Electric charge and magnetic flux are postulated to be conserved. As a consequence, the inhomogeneous and the homogeneous Maxwell equations emerge expressed in terms of the excitation H and the field strength F, respectively. H and F are assumed to fulfill a local and linear "spacetime relation" with 36 constitutive functions. The propagation of electromagnetic waves is considered under such circumstances in the geometric optics limit. We forbid birefringence in vacuum and find the light cone including its Lorentzian signature. Thus the conformally invariant part of the metric is recovered. If one sets a scale, one finds the pseudo-Riemannian metric of spacetime. 
Friedrich W. Hehl, Yuri N. Obukhov