Thermal Dynamics in General Relativity

TL;DR

This paper develops a relativistic heat conduction model using a variational multi-fluid approach, deriving a causal Cattaneo equation and connecting it with established thermodynamic models, including non-relativistic limits.

Contribution

It introduces a relativistic multi-fluid model for heat conduction that naturally incorporates causality and extends thermodynamics, comparing it with Israel-Stewart theory and deriving non-relativistic limits.

Findings

The model yields a relativistic Cattaneo equation with finite thermal relaxation time.

It shows equivalence with Israel-Stewart's second-order model at first order deviations.

The non-relativistic limit aligns with recent classical thermodynamics work.

Abstract

We discuss a relativistic model for heat conduction, building on a convective variational approach to multi-fluid systems where the entropy is treated as a distinct dynamical entity. We demonstrate how this approach leads to a relativistic version of the Cattaneo equation, encoding the finite thermal relaxation time that is required to satisfy causality. We also show that the model naturally includes the non-equilibrium Gibbs relation that is a key ingredient in most approaches to extended thermodynamics. Focussing on the pure heat conduction problem, we compare the variational results to the second-order model developed by Israel and Stewart. The comparison shows that, despite the very different philosophies behind the two approaches, the two models are equivalent at first order deviations from thermal equilibrium. Finally, we complete the picture by working out the non-relativistic limit of our results, making contact with recent work in that regime.