Conservative Moment Equations for Neutrino Radiation Transport with Limited Relativity

We derive conservative, multidimensional, energy-dependent moment equations for neutrino transport in core-collapse supernovae and related astrophysical systems, with particular attention to the consistency of conservative four-momentum and lepton number transport equations. After taking angular moments of conservative formulations of the general relativistic Boltzmann equation, we specialize to a conformally flat spacetime, which also serves as the basis for four further limits. Two of these---the multidimensional special relativistic case, and a conformally flat formulation of the spherically symmetric general relativistic case---are given in appendices for the sake of comparison with extant literature. The third limit is a weak-field, `pseudo-Newtonian' approach \citep{kim_etal_2009,kim_etal_2012} in which the source of the gravitational potential includes the trace of the stress-energy tensor (rather than just the mass density), and all orders in fluid velocity $v$ are retained. Our primary interest here is in the fourth limit: `$\mathcal{O}(v)$' moment equations for use in conjunction with Newtonian self-gravitating hydrodynamics. We show that the concept of `$\mathcal{O}(v)$' transport requires care when dealing with both conservative four-momentum and conservative lepton number transport, and present two self-consistent options: `$\mathcal{O}(v)$-plus' transport, in which an $\mathcal{O}(v^2)$ energy equation combines with an $\mathcal{O}(v)$ momentum equation to give an $\mathcal{O}(v^2)$ number equation; and `$\mathcal{O}(v)$-minus' transport, in which an $\mathcal{O}(v)$ energy equation combines with an $\mathcal{O}(1)$ momentum equation to give an $\mathcal{O}(v)$ number equation.