Relativistic Fluid Dynamics: Physics for Many Different Scales

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.

Figure 1

A “timeline” focussed on the topics covered in this review, including chemists, engineers, mathematicians, philosophers, and physicists who have contributed to the development of non-relativistic fluids, their relativistic counterparts, multi-fluid versions of both, and exotic phenomena such as superfluidity.

Figure 2

A schematic illustration of two possible versions of parallel transport. In the first case (a) a vector is transported along great circles on the sphere locally maintaining the same angle with the path. If the contour is closed, the final orientation of the vector will differ from the original one. In case (b) the sphere is considered to be embedded in a three-dimensional Euclidean space, and the vector on the sphere results from projection. In this case, the vector returns to the original orientation for a closed contour.

Figure 3

A schematic illustration of the Lie derivative. The coordinate system is dragged along with the flow, and one can imagine an observer “taking derivatives” as he/she moves with the flow (see the discussion in the text).

Figure 4

The projections at point P of a vector V^μ onto the worldline defined by U^μ and into the perpendicular hypersurface (obtained from the action of ).

Figure 5

An object with a characteristic size D is modeled as a fluid that contains M fluid elements. From inside the object we magnify a generic fluid element of characteristic size L. In order for the fluid model to work we require M ≫ N ≫ 1 and D ≫ L.

Figure 6

A local, geometrical view of the Euler equation as an integrability condition of the vorticity for a single-constituent perfect fluid.

Figure 7

The push-forward from “fluid-particle” points in the three-dimensional matter space labelled by the coordinates {X¹, X², X³} to fluid-element worldlines in spacetime. Here, the push-forward of the “I^th” (I = 1, 2, …, n) fluid-particle to, say, an initial point on a worldline in spacetime can be taken as where is the spatial position of the intersection of the worldline with the t = 0 time slice.

Figure 8

The push-forward from a point in the x^th-constituent’s three-dimensional matter space (on the left) to the corresponding “fluid-particle” worldline in spacetime (on the right). The points in matter space are labelled by the coordinates , and the constituent index x = n, s. There exist as many matter spaces as there are dynamically independent fluids, which for this case means two.