The General Relativistic Equations of Radiation Hydrodynamics in the Viscous Limit

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 fourvelocity, 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.

The Radiation Hydrodynamics of Relativistic Shear Flows

“…(84) Comparing this expression to equation (37) of Coughlin & Begelman (2014b), we see that this approach to analyzing relativistic shear is in agreement with the viscous equations of radiation hydrodynamics for divergenceless flow (∇ µ U µ = 0) if we adopt τ = 10/9.…”

“…In this paper we analyzed how a radiation field responds to regions of intense, relativistic shear, which likely arise in extreme astrophysical environments such as collapsars (MacFadyen & Woosley 1999) and tidal disruption events (Coughlin & Begelman 2014a). We considered a two-dimensional, planar shear flow, with the motion along the z-direction and the variation in that motion along the y-direction, in which the fluid appears identical at every comoving point.…”

“…In this problem, the velocity of the fluid is primarily along the z-direction and the variation in the properties of the fluid occurs predominantly along the y-direction, meaning that the geometry of the problem is identical to that established in the preceding sections. Following the procedure outlined in Coughlin & Begelman (2015a), we define the four-velocity in the z-direction as…”

“…Some progress has been made on this question from the optically thick standpoint by assuming that the radiation field is approximately isotropic in the instantaneous rest frame of the fluid. The anisotropic contribution can then be determined from the Boltzmann equation, yielding the general relativistic equations of radiation hydrodynamics in the viscous limit (Coughlin & Begelman 2014b). These equations are "viscous" in the sense that the shear of the fluid -the change in the fluid velocity over the mean free path of a photon -serves to transfer energy and momentum between the radiation and the scatterers.…”

“…An advantage of M1 is that it is easily extended to incorporate all of the effects of general and special relativity (Sadowski et al 2013). However, M1 breaks down when the radiation field is inherently very anisotropic (for example, along the axis of the jet as it propagates alongside the envelope) and it does not reduce correctly to the viscous limit (one must apply an additional, viscous term to the stress tensor in the radiation-rest frame: Coughlin & Begelman 2014b;Sadowski et al 2014).…”