  We prove that the time of classical existence of smooth solutions to the
relativistic Euler equations can be bounded from below in terms of norms that
measure the "(sound) wave-part" of the data in Sobolev space and
"transport-part" in higher regularity Sobolev space and H\"older spaces. The
solutions are allowed to have nontrivial vorticity and entropy. Our main result
is that the Sobolev norm $H^{2+}$ of the variables in the "wave-part" and the
H\"older norm $C^{0,0+}$ of the variables in the "transport-part" can be
controlled in terms of initial data for short times. We note that the Sobolev
norm assumption $H^{2+}$ is the optimal result for the variables in the
"wave-part." Compared to low regularity results for quasilinear wave equations
and the 3D non-relativistic compressible Euler equations, the main new
challenge of the paper is that when controlling the acoustic geometry and
bounding the wave equation energies, we must deal with the difficulty that the
vorticity and entropy gradient are four-dimensional space-time vectors
satisfying a space-time div-curl-transport system, where the space-time
div-curl part is not elliptic. To overcome this difficulty, we show that the
space-time div-curl systems imply elliptic div-curl-transport systems on
constant-time hypersurfaces plus error terms that involve favorable
differentiations and contractions with respect to the four-velocity. By using
these structures, we are able to adequately control the vorticity and entropy
gradient with the help of energy estimates for transport equations, elliptic
estimates, Schauder estimates, and Littlewood-Paley theory.
