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 applications across scales from laboratory experiments to cosmology, and its relevance for neutron-star modeling.

Contribution

It emphasizes the variational principle approach for relativistic fluids, offering a detailed theoretical framework distinct from standard methods, and explores its applications in astrophysics and high-energy physics.

Findings

Provides a detailed derivation of relativistic fluid equations using variational principles.

Connects the formalism to applications like neutron-star modeling and electromagnetic effects.

Highlights the differences from standard stress-energy tensor divergence methods.

Abstract

The relativistic fluid is a highly successful model used to describe the dynamics of many-particle systems moving at high velocities and/or in strong gravity. It takes as input physics from microscopic scales and yields as output predictions of bulk, macroscopic motion. By inverting the process-e.g., drawing on astrophysical observations-an understanding of relativistic features can lead to insight into physics on the microscopic scale. Relativistic fluids have been used to model systems as "small" as colliding heavy ions in laboratory experiments, 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 (multi-) fluid model. We focus on the variational principle approach championed by Brandon Carter and 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. The formalism provides a foundation for complex models, e.g., including electromagnetism, superfluidity and elasticity-all of which are relevant for state of the art neutron-star modelling.