Relativistic viscoelastic fluid mechanics

TL;DR

This paper develops a relativistic viscoelastic fluid theory that extends Navier-Stokes, remedies causality issues, and unifies various material behaviors within a covariant framework, providing a universal model for relativistic continuum materials.

Contribution

The authors construct a relativistic viscoelastic theory based on Onsager's thermodynamics that reduces to Navier-Stokes at long times and is free from acausality, unifying multiple material types.

Findings

The theory recovers relativistic Navier-Stokes in the long time limit.

Wave equations become symmetric hyperbolic, resolving acausality.

The model encompasses elastic, Maxwell, Kelvin-Voigt, and Israel-Stewart fluids.

Abstract

A detailed study is carried out for the relativistic theory of viscoelasticity which was recently constructed on the basis of Onsager's linear nonequilibrium thermodynamics. After rederiving the theory using a local argument with the entropy current, we show that this theory universally reduces to the standard relativistic Navier-Stokes fluid mechanics in the long time limit. Since effects of elasticity are taken into account, the dynamics at short time scales is modified from that given by the Navier-Stokes equations, so that acausal problems intrinsic to relativistic Navier-Stokes fluids are significantly remedied. We in particular show that the wave equations for the propagation of disturbance around a hydrostatic equilibrium in Minkowski spacetime become symmetric hyperbolic for some range of parameters, so that the model is free of acausality problems. This observation suggests that the relativistic viscoelastic model with such parameters can be regarded as a causal completion of relativistic Navier-Stokes fluid mechanics. By adjusting parameters to various values, this theory can treat a wide variety of materials including elastic materials, Maxwell materials, Kelvin-Voigt materials, and (a nonlinearly generalized version of) simplified Israel-Stewart fluids, and thus we expect the theory to be the most universal description of single-component relativistic continuum materials. We also show that the presence of strains and the corresponding change in temperature are naturally unified through the Tolman law in a generally covariant description of continuum mechanics.