Conservative form of Boltzmann's equation in general relativity

Radiation fields and their interaction with matter fields often play a crucial role in general-relativistic astrophysical phenomena; e.g., the neutrino cooling and heating play a special role in core-collapse supernova. The reason is that general-relativistic phenomena are usually accompanied by the high-density and high-temperature matter with which neutrinos strongly interact. For a physical simulation in numerical relativity, we are often required to take into account the neutrino radiation transfer effects. For this, it is necessary to solve radiation transfer equations. For strictly handling radiation transfer effects, it is necessary to numerically solve Boltzmann’s equation, taking into account the absorption, emission, and scattering source terms. This equation has a 3þ 3þ 1-dimensional form (three dimensions in real and momentum space, respectively, and one dimension in time); hence, the computational domain has to cover six-dimensional space for a simulation (unless we impose any spatial symmetry). It is an extremely challenging task to perform a wellresolved numerical simulation with a sufficient grid resolution for this equation, unless a high spatial symmetry such as spherical symmetry is imposed (e.g., see [1–4] for formulations based on the SN schemes, [5,6] for results in spherically symmetric and general-relativistic simulations, and also [7] for an alternative approach based on long characteristics and its application to protoneutron star evolution in full general relativity by means of a tangentray scheme). To date, no challenge has been made on this issue (but see [8] in Newtonian gravity). Indeed, even for formulations suitable for a numerical simulation, only a few attempts [9,10] have been reported, and there is no established formulation to date (see Ref. [10] for a review in this field). Cardall andMezzacappa [9] gave a conservative formulation of Boltzmann’s equation but not in a 3þ 1 form applicable to numerical-relativity simulations. A 3þ 1 formulation of Boltzmann’s equation was first derived by Cardall, Endeve and Mezzacappa [10] with laboratory frame in spacetime coordinates and fluid rest frame in momentum-space coordinates. As a consequence of adopting a fluid-rest-frame momentum-space coordinate basis, the resulting equations are rather complicated. An attempt to solve Boltzmann’s equation using spectral methods with the conformally flat approximation was done by [11], but their formulation was not in a conservative form. In this paper, we derive a more concise and general formulation for the conservative form of Boltzmann’s equation in general relativity. The paper is organized as follows. In Sec. II, after we briefly review the basics of Boltzmann’s equation, we describe its conservative form in a concise way. We also give conservative forms in black-hole spacetime with several coordinate conditions in real-spacetime and momentumspace coordinates. Section III is devoted to a summary. Throughout this paper, we employ the units in which the Planck constant, h, speed of light, c, and gravitational constant, G, are unity. Latin indices a, b, c, and d denote the abstract index while greek ones α; β; γ;. and latin ones i, j, k, and l denote the spacetime and spatial components, respectively.