The evolution equation for the energy-momentum moments of the non-equilibrium density function & The regularized relativistic third order hydrodynamics

In this work, we first derive the evolution equation for the general energy-momentum moment of $\delta f$, where $\delta f$ is the deviation from the local equilibrium phase space density. We then introduce a relativistic extension of regularized hydrodynamics developed in the non-relativistic case by Struchtrup and Torrilhon that combines the method of moments and Chapman-Enskog expansion. Hydrodynamic equations up to the third-order in gradients are then systematially 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 system of massless particles without any charge conservation, the third-order hydrodynamics is shown to be linearly stable and causal.