“speed of light” determined by properties of the medium [@Unruhb; @LV]. Quite generally, linearization of a field theory around a nontrivial background leads to an effective Lorentzian metric [@BarLibVis], whose signature is fixed by the hyperbolicity of the partial differential equations. At higher orders, Lorentz violations reappear [@Liberati; @Lin], and can serve as constraints on such models.

Problems arise, though, as soon as more than one kind of excitation can occur. In that case, distinct excitations typically have distinct Lorentz invariances, with different effective metrics and speeds of light [@BarLibVisb; @Sindonirev]. Even this is a special case; in general, the Lorentzian geometry becomes Finslerian, losing contact with the desired physics. One can sometimes recover a single metric by imposing a discrete symmetry on the fundamental fields [@Sindonib; @Girellib], but this step seems rather artificial. Note that it is *not* enough to simply claim that the excitations are all perturbations of the same medium: even in an ordinary elastic solid, longitudinal and transverse waves travel at different speeds.

One solution is to postulate that the “environment” is itself Lorentz invariant. If this environment is a spacetime, this begs the question: as in the EPS construction of section 2, invariance implies the 