  We present an analysis of the general relativistic Boltzmann equation for
radiation, appropriate to the case where particles and photons interact through
Thomson scattering, and derive the radiation energy-momentum tensor in the
diffusion limit, with viscous terms included. Contrary to relativistic
generalizations of the viscous stress tensor that appear in the literature, we
find that the stress tensor should contain a correction to the comoving energy
density proportional to the divergence of the four-velocity, as well as a
finite bulk viscosity. These modifications are consistent with the framework of
radiation hydrodynamics in the limit of large optical depth, and do not depend
on thermodynamic arguments such as the assignment of a temperature to the
zeroth-order photon distribution. We perform a perturbation analysis on our
equations and demonstrate that, as long as the wave numbers do not probe scales
smaller than the mean free path of the radiation, the viscosity contributes
only decaying, i.e., stable, corrections to the dispersion relations. The
astrophysical applications of our equations, including jets launched from
super-Eddington tidal disruption events and those from collapsars, are
discussed and will be considered further in future papers.
