On the stability problem in relativistic thermodynamics: implications of the Chapman-Enskog formalism

TL;DR

This paper examines the stability of relativistic fluid theories, showing that first-order models with heat and acceleration coupling are inconsistent with kinetic theory, and that second-order relaxation terms are necessary for proper stability.

Contribution

It demonstrates that the Maxwell-Cattaneo equation's relaxation term in relativistic fluids is second order, challenging the use of first-order theories for stability analysis.

Findings

First-order theories with heat and acceleration coupling are unstable.

Second-order relaxation terms are essential for stability.

First-order models contradict kinetic theory ordering.

Abstract

Extended theories are widely used in the literature to describe relativistic fluids. The motivation for this is mostly due to the causality issues allegedly present in the first order in the gradients theories. However, the decay of fluctuations in the system is also at stake when first order theories that couple heat with acceleration are used. This paper shows that although the introduction of the Maxwell-Cattaneo equation in the description of a simple relativistic fluid formally eliminates the generic instabilities identified by Hiscock and Lindblom in 1985, the hypothesis on the order of magnitude of the corresponding relaxation term contradicts the basic ordering in Knudsen's parameter present in the kinetic approach to hydrodynamics. It is shown that the time derivative, stabilizing term is of second order in such parameter and thus does not belong to the Navier-Stokes regime where the so-called instability arises.