Evolution equation for the energy-momentum moments of the nonequilibrium density function and regularized relativistic third-order hydrodynamics

In this work, we first derive the evolution equation for the general energy-momentum moment of $\ensuremath{\delta}f$, where $\ensuremath{\delta}f$ is the deviation from the local equilibrium phase-space density. We then introduce a relativistic extension of regularized hydrodynamics developed in the nonrelativistic case by Struchtrup and Torrilhon that judiciously mixes the method of moments and Chapman-Enskog expansion. Hydrodynamic equations up to the third-order in gradients are then systematically derived within the context of a single species system and the relaxation-time approximation. This is followed by a series of linear stability and causality analysis. For the massless particles without any charge conservation, the third-order hydrodynamics is shown to be linearly stable and causal.