Linearizing a Non-linear Formulation for General Relativistic Dissipative Fluids

TL;DR

This paper develops a formalism to linearize non-linear equations for dissipative relativistic fluids using an action-based approach, enabling comparison with traditional thermodynamic models and facilitating the development of causal, stable multi-fluid models.

Contribution

It introduces a method to linearize action-based equations for relativistic dissipative fluids and connects them with traditional thermodynamic flux models.

Findings

Constructed self-consistent equilibrium notions.

Developed first-order deviations on matter spaces.

Outlined a way to implement causal responses.

Abstract

Fully non-linear equations of motion for dissipative general relativistic multi-fluids can be obtained from an action principle involving the explicit use of lower dimensional matter spaces. More traditional strategies for incorporating dissipation-like the famous Mueller-Israel-Stewart model-are based on expansions away from equilibrium defined, in part, by the laws of thermodynamics. The goal here is to build a formalism to facilitate comparison of the action-based results with those based on the traditional approach. The first step of the process is to use the action-based approach itself to construct self-consistent notions of equilibrium. Next, first-order deviations are developed directly on the matter spaces, which motivates the latter as the natural arena for the underlying thermodynamics. Finally, we identify the dissipation terms of the action-based model with first-order "thermodynamical" fluxes, on which the traditional models are built. The description is developed in a general setting so that the formalism can be used to describe multi-fluid systems, for which causal and stable models are not yet available. As an illustration of the approach, a simple application of a single viscous fluid is considered and, even though the expansion is halted at first order, we sketch how a causal response can be implemented through Cattaneo-type equations.