The Radiative Stress Tensor

We use the transfer equation in relativistic form to develop an expansion of the onephoton distribution for a medium with constant photon mean free path, ǫ. On carrying out appropriate integrations and manipulations, we convert this expansion into one for the frequency-integrated intensity. We regroup the terms of the intensity expansion according to both the power of ǫ and the angular structure of the various terms and then carry out angle integrations to obtain the expansions for the components of the stress energy tensor: the radiative energy density, the radiative flux and the pressure tensor. In leading order, we recover Thomas’ (1930) results for the viscosity tensor and his expression for the viscosity coefficient, which are correct for short mean free paths. As had been done earlier for the radiative heat equation (Unno and Spiegel 1966), we keep at each order in the expansion a dominant portion, but this time one with a richer angular structure. Then, after some rearrangement of the various summations in the expressions for the moments, we replace the sum of the calculated higher order terms by a Padé approximant, or rational approximation (Baker 1975), to provide an improved closure approximation for the radiative stress tensor. The resulting radiative viscosity tensor may be expressed either as a simple integral operator acting on the Thomas stress tensor or as the solution of an inhomogenous, linear partial differential equation. The expression obtained for the radiative viscosity tensor applies for media with long, as well as short, photon mean free paths. We also develop results applicable for relatively smooth flows by using the form of the Thomas stress tensor with generalized transport coefficients derived by the application of a suitable operator to the bare Thomas coefficients.