The relativistic kinetic dispersion relation: Comparison of the relativistic Bhatnagar–Gross–Krook model and Grad’s 14-moment expansion

In this paper, we study the Cauchy problem of the linearized kinetic equations for the models of Marle and Anderson–Witting, and compare these dispersion relations with the 14-moment theory. First, we propose a modification of the Marle model to improve the resultant transport coefficients in accordance with those obtained by the full Boltzmann equation. Using the modified Marle model and Anderson–Witting model, we calculate dispersion relations that are kinetically correct within the validity of the BGK approximation. The 14-moment theory that includes the time derivative of dissipation currents has a causal structure, in contrast to the acausal first-order Chapman–Enskog approximation. However, the dispersion relation of the 14-moment theory does not accurately describe the result of the kinetic equation. Thus, our calculation indicates that keeping these second-order terms does not simply correspond to improving the physical description of the relativistic hydrodynamics.