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We present an analysis of the general relativistic Boltzmann equation forradiation, appropriate to the case where particles and photons interact throughThomson scattering, and derive the radiation energy-momentum tensor in thediffusion limit, with viscous terms included. Contrary to relativisticgeneralizations of the viscous stress tensor that appear in the literature, wefind that the stress tensor should contain a correction to the comoving energydensity proportional to the divergence of the four-velocity, as well as afinite bulk viscosity. These modifications are consistent with the framework ofradiation hydrodynamics in the limit of large optical depth, and do not dependon thermodynamic arguments such as the assignment of a temperature to thezeroth-order photon distribution. We perform a perturbation analysis on ourequations and demonstrate that, as long as the wave numbers do not probe scalessmaller than the mean free path of the radiation, the viscosity contributesonly decaying, i.e., stable, corrections to the dispersion relations. Theastrophysical applications of our equations, including jets launched fromsuper-Eddington tidal disruption events and those from collapsars, arediscussed and will be considered further in future papers.

The general relativistic equations of radiation hydrodynamics in the viscous limit