Stability, causality, and hyperbolicity in Carter's "regular" theory of relativistic heat-conducting fluids

Stability and causality are studied for linear perturbations about equilibrium in Carter's "regular" theory of relativistic heat-conducting fluids. The "regular" theory, when linearized around an equilibrium state having vanishing expansion and shear, is shown to be equivalent to the inviscid limit of the linearized Israel-Stewart theory of relativistic dissipative fluids for a particular choice of the second-order coefficients ${\beta }_1$ and ${\gamma }_2$. A set of stability conditions is determined for linear perturbations of a general inviscid Israel-Stewart fluid using a monotonically decreasing energy functional. It is shown that, as in the viscous case, stability implies that the characteristic velocities are subluminal and that perturbations obey hyperbolic equations. The converse theorem is also true. We then apply this analysis to a nonrelativistic Boltzmann gas and to a strongly degenerate free Fermi gas in the "regular" theory. Carter's "regular" theory is shown to be incapable of correctly describing the nonrelativistic Boltzmann gas and the degenerate Fermi gas (at all temperatures).