The general relativistic equations of radiation hydrodynamics in the viscous limit

TL;DR

This paper derives a refined set of relativistic radiation hydrodynamics equations including viscous effects, correcting previous models and analyzing their stability and astrophysical applications.

Contribution

It introduces a new form of the viscous stress tensor in relativistic radiation hydrodynamics, incorporating a correction to the energy density and finite bulk viscosity, consistent with high optical depth conditions.

Findings

Viscous corrections are stable for scales larger than the mean free path.

The stress tensor correction involves a divergence of the four-velocity.

Applications include modeling jets from tidal disruption events and collapsars.

Abstract

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.