Relativistic Fluid Dynamics: Physics for Many Different Scales

TL;DR

This review discusses the mathematical foundations and variational principle approach of relativistic fluid dynamics, highlighting its wide applicability from microscopic particles to cosmological scales.

Contribution

It emphasizes the variational principle method for deriving relativistic fluid equations, offering insights into both microscopic and macroscopic physics.

Findings

Clarifies the variational approach to relativistic fluids.

Connects microscopic physics to large-scale phenomena.

Provides detailed discussion of conservation laws and applications.

Abstract

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.