The relativistic fluid is a highly successful model used to describe the dynamics of many-particle, relativistic systems. It takes as input basic physics from microscopic scales and yields as output predictions of bulk, macroscopic motion. By inverting the process, an understanding of bulk features can lead to insight into physics on the microscopic scale. Relativistic fluids have been used to model systems as “small” as heavy ions in collisions, and as large as the Universe itself, with “intermediate” sized objects like neutron stars being considered along the way. The purpose of this review is to discuss the mathematical and theoretical physics underpinnings of the relativistic (multiple) fluid model. We focus on the variational principle approach championed by Brandon Carter and his collaborators, in which a crucial element is to distinguish the momenta that are conjugate to the particle number density currents. This approach differs from the “standard” text-book derivation of the equations of motion from the divergence of the stress-energy tensor in that one explicitly obtains the relativistic Euler equation as an “integrability” condition on the relativistic vorticity. We discuss the conservation laws and the equations of motion in detail, and provide a number of (in our opinion) interesting and relevant applications of the general theory.

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Relativistic fluid dynamics: physics for many different scales

We review the progress made in extended irreversible thermodynamics during the ten years that have elapsed since the publication of our first review on the same subject (Rep. Frog. Phys. 1988 51 1105-72). During this decade much effort has been devoted to achieving a better understanding of the fundamentals and a broadening of the domain of applications. The macroscopic formulation of extended irreversible thermodynamics is reviewed and compared with other non-equilibrium thermodynamic theories. The foundations of EIT are discussed on the bases of information theory, kinetic theory, stochastic phenomena and computer simulations. Several significant applications are presented, some of them of considerable practical interest (non-classical heat transport, polymer solutions, non-Fickian diffusion, microelectronic devices, dielectric relaxation), and some others of special theoretical appeal (superfluids, nuclear collisions, cosmology). We also outline some basic problems which are not yet completely solved, such as the definitions of entropy and temperature out of equilibrium, the selection of the relevant variables, and the status to be reserved to the H-theorem and its relation to the second law. In writing this review, we had four objectives in mind: to show (i) that extended irreversible thermodynamics stands at the frontiers of modern thermodynamics; (ii) that it opens the way to new and useful applications; (iii) that much progress has been achieved during the last decade, and (iv) that the subject is far from being exhausted.

https://iopscience.iop.org/article/10.1088/0034-4885/62/7/201/pdf

Extended irreversible thermodynamics revisited (1988-98)

Applying the 1 + 3 formalism we write down the full set of equations governing the structure and the evolution of self-gravitating cylindrically symmetric dissipative fluids with anisotropic stresses, in terms of scalar quantities obtained from the orthogonal splitting of the Riemann tensor (structure scalars), in the context of general relativity. These scalars which have been shown previously (in the spherically symmetric case) to be related to fundamental properties of the fluid distribution, such as: energy density, energy density inhomogeneity, local anisotropy of pressure, dissipative flux, active gravitational mass etc, are shown here to play also a very important role in the dynamics of cylindrically symmetric fluids. It is also shown that in the static case, all possible solutions to Einstein equations may be expressed explicitly through three of these scalars.

Cylindrically symmetric relativistic fluids: a study based on structure scalars

We find a simple exact model of radiating stellar collapse, with a shear-free and non-accelerating interior matched to a Vaidya exterior. The heat flux is subject to causal thermodynamics, leading to self-consistent determination of the temperature T. We solve for T exactly when the mean collision time is constant, and perturbatively in a more realistic case of variable . Causal thermodynamics predicts temperature behaviour that can differ significantly from the predictions of non-causal theory. In particular, the causal theory gives a higher central temperature and greater temperature gradient.

A causal model of radiating stellar collapse

We present some analytical solutions to the Einstein equations, describing radiating collapsing spheres in the diffusion approximation. Solutions allow for modeling physical reasonable situations. The temperature is calculated for each solution, using a hyperbolic transport equation, which permits to exhibit the influence of relaxational effects on the dynamics of the system.

Some analytical models of radiating collapsing spheres

We study, for spherically symmetric fluid distributions undergoing dissipation in the form of a radial heat flow, the role played by the relaxation time in a diffusion process. We show that the temperature gradient appearing as a result of perturbations, and thereby the subsequent evolution of the system, is highly dependent on the product of relaxation time by the period of oscillation of the· sphere.

N. O. Santos

Papers

Thermal evolution of compact objects and relaxation time