Symmetric hyperbolic form of central-moment relativistic gas dynamics with gravitation

Recently, Banach and Larecki (2002 Rev. Math. Phys. 14 469) systematically derived the evolution equations for a gas obtained by taking moments of the relativistic Boltzmann equation, truncating and closing the resulting system of moment equations by a maximum-entropy ansatz. The moments just mentioned (so-called central moments) are defined with respect to a network of fiducial observers whose 4-velocities are at first arbitrary (though fixed ab initio) and later specialized to be nonrotating (hypersurface-orthogonal). The basic advantage of the latter choice is that if one expresses the central moments and the collision operator moments in terms of Lagrange multipliers, the evolution equations for these multipliers are then automatically symmetric hyperbolic and causal at every order of truncation. In this paper, the formulation of central-moment relativistic gas dynamics goes further by combining it with the dynamics of gravitational fields. Explicitly, we employ the 1 + 3 orthonormal frame formalism for spacetime, which contains the equations for the tetrad basis and the Ricci rotation coefficients, to derive a full causal evolution system of first-order symmetric hyperbolic form for a set of Lagrange multipliers and gravitational fields. An important issue arising in our studies is the question of consistency of equations of two different kinds: constraint equations and evolution equations. Thus we show that these two sets of equations are consistent with each other, in the sense that the constraint equations are preserved by the evolution equations.