[Paper Review] Covariant Thermodynamics and Relativity

C. S. López-Monsalvo|arXiv (Cornell University)|Jul 6, 2011

Gas Dynamics and Kinetic Theory4 references3 citations

TL;DR

This paper proposes a covariant, variational formulation of relativistic heat conduction using a two-fluid model of matter and entropy, deriving a causal generalization of the Cattaneo equation that resolves the acausality of the Eckart model. The approach naturally introduces a relaxation term via Lie differentiation along the fluid flow, providing a thermodynamically consistent, causal theory of heat transport without ad-hoc parameters.

ABSTRACT

This thesis deals with the dynamics of irreversible processes within the context of the general theory of relativity. In particular, we address the problem of the 'infinite' speed of propagation of thermal disturbances in a dissipative fluid. The present work builds on the multi-fluid variational approach to relativistic dissipation, pioneered by Carter, and provides a dynamical theory of heat conduction. The novel property of such approach is the thermodynamic interpretation associated with a two-fluid system whose constituents are matter and entropy. The dynamics of this model leads to a relativistic generalisation of the Cattaneo equation; the constitutive relation for causal heat transport. A comparison with the Israel and Stewart model is presented and its equivalence is shown. This discussion provides new insights into the not-well understood definition of a non-equilibrium temperature. The variational approach to heat conduction presented in this thesis constitutes a mathematically promising formalism to explore the relativistic evolution towards equilibrium of dissipative fluids in a dynamical manner and to get a deeper conceptual understanding of non-equilibrium thermodynamic quantities. Moreover, it might also be useful to explore the more fundamental issues of the irreversible dynamics of relativity and its connections with the time asymmetry of nature.

Motivation & Objective

• To resolve the causality paradox in relativistic heat conduction, where the Eckart model predicts infinite signal propagation speeds.
• To develop a thermodynamically consistent, relativistic generalization of the Cattaneo equation using a variational multi-fluid approach.
• To provide a fundamental, rather than phenomenological, derivation of the relaxation term in heat flux dynamics.
• To analyze the stability and causality of relativistic dissipative systems using a two-stream instability framework in the relativistic domain.
• To clarify the physical meaning of non-equilibrium temperature and the role of effective inertia in heat flux dynamics.

Proposed method

• Formulates a two-fluid system with matter and entropy as distinct fluid components, each with their own 4-velocities and conserved currents.
• Applies a variational principle to the Lagrangian density of the system, deriving equations of motion via the Euler-Lagrange equations.
• Introduces the Lie derivative $\mathcal{L}_{\rm u}(q^a/s)$ along the fundamental flow vector $u^a$ to model the relaxation of the heat flux.
• Derives a relativistic generalization of the Cattaneo equation as the constitutive relation for causal heat transport.
• Analyzes the two-stream instability in the relativistic regime, assessing stability and causality directly from the equations of state or Lagrangian densities.
• Compares the variational model with the Israel-Stewart second-order theory, demonstrating equivalence under specific conditions.

Experimental results

Research questions

• RQ1 How can a relativistic theory of heat conduction be formulated in a way that ensures causality and thermodynamic consistency?
• RQ2 What is the physical origin of the relaxation term in the Cattaneo-type equation within a relativistic framework?
• RQ3 How does the effective inertia of the heat flux, interpreted as mass per unit entropy, affect the dynamics of heat transport?
• RQ4 In what conditions does the regularity condition (inertia proportional to temperature) hold, and when do deviations (e.g., in superfluids) lead to measurable effects?
• RQ5 Can the two-stream instability analysis be meaningfully extended to the relativistic domain to assess the stability of dissipative fluid models?

Key findings

• The variational two-fluid model naturally produces a relativistic Cattaneo-type equation with a relaxation term $\mathcal{L}_{\rm u}(q^a/s)$, ensuring causal heat propagation.
• The theory resolves the acausality of the Eckart model by deriving the relaxation term from fundamental thermodynamic principles rather than ad-hoc assumptions.
• The effective inertia of the heat flux is shown to be proportional to the temperature $\theta$, corresponding to the rest mass of energy per unit entropy.
• The model is shown to be equivalent to the Israel-Stewart second-order theory under appropriate conditions, validating its consistency with established frameworks.
• The two-stream instability analysis is extended to the relativistic domain, allowing direct assessment of stability and causality from the Lagrangian or equation of state.
• Deviations from regularity (e.g., in superfluids) are shown to lead to significant anomalies in effective inertia, analogous to anomalous magnetic moments in particle physics.