  We derive a new formulation of the relativistic Euler equations that exhibits
remarkable properties. This new formulation consists of a coupled system of
geometric wave, transport, and elliptic equations, sourced by nonlinearities
that are null forms relative to the acoustical metric. Our new formulation is
well-suited for various applications, in particular for the study of stable
shock formation, as it is surveyed in the paper. Moreover, using the new
formulation presented here, we establish a local well-posedness result showing
that the vorticity and the entropy of the fluid are one degree more
differentiable compared to the regularity guaranteed by standard estimates
(assuming that the initial data enjoy the extra differentiability). This gain
in regularity is essential for the study of shock formation without symmetry
assumptions. Our results hold for an arbitrary equation of state, not
necessarily of barotropic type.
