  We develop first-principles theory of relativistic fluid turbulence at high
Reynolds and P\'eclet numbers. We follow an exact approach pioneered by
Onsager, which we explain as a non-perturbative application of the principle of
renormalization-group invariance. We obtain results very similar to those for
non-relativistic turbulence, with hydrodynamic fields in the inertial-range
described as distributional or "coarse-grained" solutions of the relativistic
Euler equations. These solutions do not, however, satisfy the naive
conservation-laws of smooth Euler solutions but are afflicted with dissipative
anomalies in the balance equations of internal energy and entropy. The
anomalies are shown to be possible by exactly two mechanisms, local cascade and
pressure-work defect. We derive "4/5th-law"-type expressions for the anomalies,
which allow us to characterize the singularities (structure-function scaling
exponents) required for their non-vanishing. We also investigate the Lorentz
covariance of the inertial-range fluxes, which we find is broken by our
coarse-graining regularization but which is restored in the limit that the
regularization is removed, similar to relativistic lattice quantum field
theory. In the formal limit as speed of light goes to infinity, we recover the
results of previous non-relativistic theory. In particular, anomalous heat
input to relativistic internal energy coincides in that limit with anomalous
dissipation of non-relativistic kinetic energy.
