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.

相对论流体是一个非常成功的模型，用于描述以高速和/或强重力运动的多粒子系统的动力学。它将微观尺度的物理输入作为输入，并将产量作为体积、宏观运动的输出预测。通过反转这个过程——例如，利用天体物理学观察——对相对论特征的理解可以导致对微观尺度物理学的洞察。相对论流体已被用于模拟系统，如实验室实验中碰撞重离子的“小”系统，以及与宇宙本身一样大的系统，沿途考虑“中等”大小的物体，如中子星。本综述的目的是讨论相对论（多）流体模型的数学和理论物理基础。我们专注于 Brandon Carter 和合作者倡导的变分原理方法，其中一个关键要素是区分与粒子数密度电流共轭的动量。这种方法不同于“标准”教科书从应力-能量张量的散度推导运动方程，因为它明确地获得了相对论欧拉方程作为相对论涡度的“可积性”条件。我们详细讨论了守恒定律和运动方程，并提供了（在我们看来）一般理论的一些有趣和相关的应用。形式主义为复杂模型提供了基础，例如，包括电磁学、超流性和弹性——所有这些都与最先进的中子星建模相关。