The regular conducting fluid model for relativistic thermodynamics

TL;DR

This paper introduces a new 'regular' relativistic fluid model that ensures physically realistic, hyperbolic behavior and subluminal signal propagation, improving upon previous models like Eckart and Landau-Lifshitz.

Contribution

It presents a modern, mathematically consistent relativistic conducting fluid model derived from a variational approach, addressing limitations of earlier models.

Findings

The new model is strictly hyperbolic and causally well-behaved.

It aligns with physical requirements like subluminal signal speeds.

It offers a more natural and mathematically sound framework for relativistic thermodynamics.

Abstract

The "regular" model presented here can be considered to be the most natural solution to the problem of constructing the simplest possible relativistic analogue of the category of classical Fourier--Euler thermally conducting fluid models as characterised by a pair of equations of state for just two dependent variables (an equilibrium density and a conducting scalar). The historically established but causally unsatisfactory solution to this problem due to Eckart is shown to be based on an ansatz that is interpretable as postulating a most unnatural relation between the (particle and entropy) velocities and their associated momenta, which accounts for the well known bad behaviour of that model which has recently been shown to have very pathological mixed-elliptic-hyperbolic comportments. The newer (and more elegant) solution of Landau and Lifshitz has a more mathematically respectable parabolic-hyperbolic comportment, but is still compatible with a well posed initial value problem only in such a restricted limit-case such as that of linearised perturbations of a static background. For mathematically acceptable behaviour undermore general circumstances, and a fortiori for the physically motivated requirement of subluminal signal propagation, only strictly hyperbolic behaviour is acceptable. Attention is drawn here to the availability of a more modern "regular" solution which, is fully satisfactory as far as all these requirements are concerned. This "regular" category of relativistic conducting fluid models arises naturally within a recently developed variational approach, in which the traditionally important stress--momentum-energy density tensor is relegated to a secondary role, while the relevant covariant 4-momentum co-vectors are instead brought to the fore.