The Merriam-Webster online dictionary (http://www.m-w.com/) defines a fluid as “…a substance (as a liquid or gas) tending to flow or conform to the outline of its container” when taken as a noun and “…having particles that easily move and change their relative position without a separation of the mass and that easily yield to pressure: capable of flowing” when taken as an adjective. The best model of physics is the Standard Model which is ultimately the description of the “substance” that will make up our fluids. The substance of the Standard Model consists of remarkably few classes of elementary particles: leptons, quarks, and so-called “force” carriers (gauge-vector bosons). Each elementary particle is quantum mechanical, but the Einstein equations require explicit trajectories. Moreover, cosmology and neutron stars are basically many particle systems and, even forgetting about quantum mechanics, it is not practical to track each and every “particle” that makes them up, regardless of whether these are elementary (leptons, quarks, etc.) or collections of elementary particles (e.g. stars in galaxies and galaxies in cosmology). The fluid model is such that the inherent quantum mechanical behavior, and the existence of many particles are averaged over in such a way that it can be implemented consistently in the Einstein equations.

Central to the model is the notion of a “fluid particle”, also known as a “fluid element” or “material particle” [68]. It is an imaginary, local “box” that is infinitesimal with respect to the system en masse and yet large enough to contain a large number of particles (e.g. an Avogadro’s number of particles). This is illustrated in Figure 5. We consider an object with characteristic size that is modeled as a fluid that contains fluid elements. From inside the object we magnify a generic fluid element of characteristic size . In order for the fluid model to work we require and . Strictly speaking, our model has infinitesimal, , but with the total number of particles remaining finite. An operational point of view is that discussed by Lautrup in his fine text “Physics of Continuous Matter” [68]. He rightly points out that implicit in the model is some statement of the intended precision. At some level, any real system will be discrete and no longer represented by a continuum. As long as the scale where the discreteness of matter and fluctuations are important is much smaller than the desired precision, a continuum approximation is valid.

The explicit trajectories that enter the Einstein equations are those of the fluid elements, not the much smaller (generally fundamental) particles that are “confined”, on average, to the elements. Hence, when we speak later of the fluid velocity, we mean the velocity of fluid elements. In this sense, the use of the phrase “fluid particle” is very apt. For instance, each fluid element will trace out a timelike trajectory in spacetime. This is illustrated in Figure 7 for a number of fluid elements. An object like a neutron star is a collection of worldlines that fill out continuously a portion of spacetime. In fact, we will see later that the relativistic Euler equation is little more than an “integrability” condition that guarantees that this filling (or fibration) of spacetime can be performed. The dual picture to this is to consider the family of three-dimensional hypersurfaces that are pierced by the worldlines at given instants of time, as illustrated in Figure 7. The integrability condition in this case will guarantee that the family of hypersurfaces continuously fill out a portion of spacetime. In this view, a fluid is a so-called three-brane (see [21] for a general discussion of branes). In fact the method used in Section 8 to derive the relativistic fluid equations is based on thinking of a fluid as living in a three-dimensional “matter” space (i.e. the left-hand-side of Figure 7).

Once one understands how to build a fluid model using the matter space, it is straight-forward to extend the technique to single fluids with several constituents, as in Section 9, and multiple fluid systems, as in Section 10. An example of the former would be a fluid with one species of particles at a non-zero temperature, i.e. non-zero entropy, that does not allow for heat conduction relative to the particles. (Of course, entropy does flow through spacetime.) The latter example can be obtained by relaxing the constraint of no heat conduction. In this case the particles and the entropy are both considered to be fluids that are dynamically independent, meaning that the entropy will have a four-velocity that is generally different from that of the particles. There is thus an associated collection of fluid elements for the particles and another for the entropy. At each point of spacetime that the system occupies there will be two fluid elements, in other words, there are two matter spaces (cf. Section 10). Perhaps the most important consequence of this is that there can be a relative flow of the entropy with respect to the particles. In general, relative flows lead to the so-called entrainment effect, i.e. the momentum of one fluid in a multiple fluid system is in principle a linear combination of all the fluid velocities [6]. The canonical examples of two fluid models with entrainment are superfluid [94] at non-zero temperature and a mixture of superfluid and [8].

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