Riemannian light cone from vanishing birefringence in premetric vacuum electrodynamics

We consider premetric electrodynamics with a local and linear constitutive law for the vacuum. Within this framework, we find quartic Fresnel wave surfaces for the propagation of light. If we require (i) the Fresnel equation to have only real solutions and (ii) the vanishing of birefringence in vacuum, then a Riemannian light cone is implied. No proper Finslerian structure can occur. This is generalized to dynamical equations of any order.