Steps for Solving the Transfer Equation for Arbitrary Flows in Stationary Spacetimes

We derive the radiative transfer equation for arbitrary stationary relativistic flows in stationary spacetimes, i.e. for steady-state transfer problems. We show how the standard characteristics method of solution developed by Mihalas and used throughout the radiative transfer community can be adapted to multi-dimensional applications with isotropic sources. Because the characteristics always coincide with geodesics and can always be specified by constants, direct integration of the characteristics derived from the transfer equation as commonly done in 1-D applications is not required. The characteristics are known for a specified metric from the geodesics. We give details in both flat and static spherically symmetric spacetimes. This work has direct application in 3-dimensional simulations of supernovae, gamma-ray bursts, and active galactic nuclei, as well as in modeling neutron star atmospheres.