For instance, whenever one has a system of PDEs that can be written in first-order quasi-linear symmetric hyperbolic
form, then it is an exact non-perturbative result that the matrix of coefficients for the first-derivative terms can
be used to construct a conformal class of metrics that encodes the causal structure of the system of PDEs.
For barotropic hydrodynamics this is briefly discussed in [138]. This analysis is related to the behaviour of
characteristics of the PDEs, and ultimately can be linked back to the Fresnel equation that appears in the eikonal
limit.