Relativistic Fluid Dynamics: Physics for Many Different Scales
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Abstract
The relativistic fluid is a highly successful model used to describe the dynamics of manyparticle, 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” textbook derivation of the equations of motion from the divergence of the stressenergy 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.
Keywords
Neutron Star Covariant Derivative Parallel Transport Particle Number Density Fluid Element1 Introduction
1.1 Setting the stage
If one does a search on the topic of relativistic fluids on any of the major physics article databases one is overwhelmed by the number of “hits”. This is a manifestation of the importance that the fluid model has long had for modern physics and engineering. For relativistic physics, in particular, the fluid model is essential. After all, manyparticle astrophysical and cosmological systems are the best sources of detectable effects associated with General Relativity. Two obvious examples, the expansion of the Universe and oscillations of neutron stars, indicate the vast range of scales on which relativistic fluid models are relevant. A particularly exciting context for general relativistic fluids today is their use in the modeling of sources of gravitational waves. This includes the compact binary inspiral problem, either of two neutron stars or a neutron star and a black hole, the collapse of stellar cores during supernovae, or various neutron star instabilities. One should also not forget the use of special relativistic fluids in modeling collisions of heavy nuclei, astrophysical jets, and gammaray burst sources.
This review provides an introduction to the modeling of fluids in General Relativity. Our main target audience is graduate students with a need for an understanding of relativistic fluid dynamics. Hence, we have made an effort to keep the presentation pedagogical. The article will (hopefully) also be useful to researchers who work in areas outside of General Relativity and gravitation per se (e.g. a nuclear physicist who develops neutron star equations of state), but who require a working knowledge of relativistic fluid dynamics.
Throughout most of the article we will assume that General Relativity is the proper description of gravity. Although not too severe, this is a restriction since the problem of fluids in other theories of gravity has interesting aspects. As we hope that the article will be used by students and researchers who are not necessarily experts in General Relativity and techniques of differential geometry, we have included some discussion of the mathematical tools required to build models of relativistic objects. Even though our summary is not a proper introduction to General Relativity we have tried to define the tools that are required for the discussion that follows. Hopefully our description is sufficiently selfcontained to provide a less experienced reader with a working understanding of (at least some of) the mathematics involved. In particular, the reader will find an extended discussion of the covariant and Lie derivatives. This is natural since many important properties of fluids, both relativistic and nonrelativistic, can be established and understood by the use of parallel transport and Liedragging. But it is vital to appreciate the distinctions between the two.
Ideally, the reader should have some familiarity with standard fluid dynamics, e.g. at the level of the discussion in Landau and Lifshitz [66], basic thermodynamics [96], and the mathematics of action principles and how they are used to generate equations of motion [65]. Having stated this, it is clear that we are facing a real challenge. We are trying to introduce a topic on which numerous books have been written (e.g. [112, 66, 72, 9, 123]), and which requires an understanding of much of theoretical physics. Yet, one can argue that an article of this kind is timely. In particular, there have recently been exciting developments for multiconstituent systems, such as superfluid/superconducting neutron star cores^{1}. Much of this work has been guided by the geometric approach to fluid dynamics championed by Carter [17, 19, 21]. This provides a powerful framework that makes extensions to multifluid situations quite intuitive. A typical example of a phenomenon that arises naturally is the socalled entrainment effect, which plays a crucial role in a superfluid neutron star core. Given the potential for future applications of this formalism, we have opted to base much of our description on the work of Carter and his colleagues.
Even though the subject of relativistic fluids is far from new, a number of issues remain to be resolved. The most obvious shortcoming of the present theory concerns dissipative effects. As we will discuss, dissipative effects are (at least in principle) easy to incorporate in Newtonian theory but the extension to General Relativity is problematic (see, for instance, Hiscock and Lindblom [54]). Following early work by Eckart [39], a significant effort was made by Israel and Stewart [58, 57] and Carter [17, 19]. Incorporation of dissipation is still an active enterprise, and of key importance for future gravitationalwave asteroseismology which requires detailed estimates of the role of viscosity in suppressing possible instabilities.
1.2 A brief history of fluids
The two fluids air and water are essential to human survival. This obvious fact implies a basic human need to divine their innermost secrets. Homo Sapiens has always needed to anticipate air and water behavior under a myriad of circumstances, such as those that concern water supply, weather, and travel. The essential importance of fluids for survival, and that they can be exploited to enhance survival, implies that the study of fluids probably reaches as far back into antiquity as the human race itself. Unfortunately, our historical record of this everongoing study is not so great that we can reach very far accurately.
A wonderful account (now in Dover print) is “A History and Philosophy of Fluid Mechanics” by G.A. Tokaty [111]. He points out that while early cultures may not have had universities, government sponsored labs, or privately funded centers pursuing fluids research (nor the Living Reviews archive on which to communicate results!), there was certainly some collective understanding. After all, there is a clear connection between the viability of early civilizations and their access to water. For example, we have the societies associated with the Yellow and Yangtze rivers in China, the Ganges in India, the Volga in Russia, the Thames in England, and the Seine in France, to name just a few. We should also not forget the Babylonians and their amazing technological (irrigation) achievements in the land between the Tigris and Euphrates, and the Egyptians, whose intimacy with the flooding of the Nile is well documented. In North America, we have the socalled Mississippians, who left behind their moundbuilding accomplishments. For example, the Cahokians (in Collinsville, Illinois) constructed Monk’s Mound, the largest preColumbian earthen structure in existence that is “…over 100 feet tall, 1000 feet long, and 800 feet wide (larger at its base than the Great Pyramid of Giza)” (see http://en.wikipedia.org/wiki/Monk’s_Mound).
In terms of ocean and sea travel, we know that the maritime ability of the Mediterranean people was a main mechanism for ensuring cultural and economic growth and societal stability. The finelytuned skills of the Polynesians in the South Pacific allowed them to travel great distances, perhaps reaching as far as South America, and certainly making it to the “most remote spot on the Earth”, Easter Island. Apparently, they were remarkably adept at reading the smallest of signs — water color, views of weather on the horizon, subtleties of wind patterns, floating objects, birds, etc. — as indication of nearby land masses. Finally, the harsh climate of the North Atlantic was overcome by the highly accomplished Nordic sailors, whose skills allowed them to reach several sites in North America. Perhaps it would be appropriate to think of these early explorers as adept geophysical fluid dynamicists/oceanographers?
Tokaty [111] discusses the human propensity for destruction when it comes to our water resources. Depletion and pollution are the main offenders. He refers to a “Battle of the Fluids” as a struggle between their destruction and protection. His context for this discussion was the Cold War. He rightly points out the failure to protect our water and air resources by the two predominant participants — the USA and USSR. In an ironic twist, modern study of the relativistic properties of fluids has also a “Battle of the Fluids”. A selfgravitating mass can become absolutely unstable and collapse to a black hole, the ultimate destruction of any form of matter.
1.3 Notation and conventions
Throughout the article we assume the “MTW” [80] conventions. We also generally assume geometrized units c = G = 1, unless specifically noted otherwise. A coordinate basis will always be used, with spacetime indices denoted by lowercase Greek letters that range over {0, 1, 2, 3} (time being the zeroth coordinate), and purely spatial indices denoted by lowercase Latin letters that range over {1, 2, 3}. Unless otherwise noted, we assume the Einstein summation convention.
2 Physics in a Curved Spacetime
There is an extensive literature on special and general relativity, and the spacetimebased view^{2} of the laws of physics. For the student at any level interested in developing a working understanding we recommend Taylor and Wheeler [109] for an introduction, followed by Hartle’s excellent recent text [51] designed for students at the undergraduate level. For the more advanced students, we suggest two of the classics, “MTW” [80] and Weinberg [117], or the more contemporary book by Wald [114]. Finally, let us not forget the Living Reviews archive as a premier online source of uptodate information!
In terms of the experimental and/or observational support for special and general relativity, we recommend two articles by Will that were written for the 2005 World Year of Physics celebration [122, 121]. They summarize a variety of tests that have been designed to expose breakdowns in both theories. (We also recommend Will’s popular book Was Einstein Right? [119] and his technical exposition Theory and Experiment in Gravitational Physics [120].) To date, Einstein’s theoretical edifice is still standing!
For special relativity, this is not surprising, given its long list of successes: explanation of the MichaelsonMorley result, the prediction and subsequent discovery of antimatter, and the standard model of particle physics, to name a few. Will [122] offers the observation that genetic mutations via cosmic rays require special relativity, since otherwise muons would decay before making it to the surface of the Earth. On a more somber note, we may consider the Trinity site in New Mexico, and the tragedies of Hiroshima and Nagasaki, as reminders of E = mc^{2}.
In support of general relativity, there are Eötvöstype experiments testing the equivalence of inertial and gravitational mass, detection of gravitational redshifts of photons, the passing of the solar system tests, confirmation of energy loss via gravitational radiation in the HulseTaylor binary pulsar, and the expansion of the Universe. Incredibly, general relativity even finds a practical application in the GPS system: If general relativity is neglected, an error of about 15 meters results when trying to resolve the location of an object [122]. Definitely enough to make driving dangerous!

test bodies fall with the same acceleration independently of their internal structure or composition;

the outcome of any local nongravitational experiment is independent of the velocity of the freelyfalling reference frame in which it is performed;

the outcome of any local nongravitational experiment is independent of where and when in the Universe it is performed.
 1.
spacetime is endowed with a symmetric metric,
 2.
the trajectories of freely falling bodies are geodesics of that metric, an
 3.
in local freely falling reference frames, the nongravitational laws of physics are those of special relativity.
2.1 The metric and spacetime curvature
Our strategy is to provide a “working understanding” of the mathematical objects that enter the Einstein equations of general relativity. We assume that the metric is the fundamental “field” of gravity. For fourdimensional spacetime it determines the distance between two spacetime points along a given curve, which can generally be written as a one parameter function with, say, components x^{ μ }(λ). As we will see, once a notion of parallel transport is established, the metric also encodes information about the curvature of its spacetime, which is taken to be of the pseudoRiemannian type, meaning that the signature of the metric is − + ++ (cf. Equation (2) below).
2.2 Parallel transport and the covariant derivative
For both physical and mathematical reasons, one expects a covariant derivative to be defined in terms of a limit. This is, however, a bit problematic. In threedimensional Euclidean space limits can be defined uniquely in that vectors can be moved around without their lengths and directions changing, for instance, via the use of Cartesian coordinates (the {i, j, k} set of basis vectors) and the usual dot product. Given these limits those corresponding to more general curvilinear coordinates can be established. The same is not true for curved spaces and/or spacetimes because they do not have an a priori notion of parallel transport.
On the other hand, we could consider that the sphere is embedded in a threedimensional Euclidean space, and that the twodimensional vector on the sphere results from projection of a threedimensional vector. Move the projection so that its higherdimensional counterpart always maintains the same orientation with respect to its original direction in the embedding space. When the projection returns to its starting place it will have exactly the same orientation as it started with (see Figure 2). It is thus clear that a derivative operation that depends on comparing a vector at one point to that of a nearby point is not unique, because it depends on the choice of parallel transport.
Pauli [88] notes that LeviCivita [71] is the first to have formulated the concept of parallel “displacement”, with Weyl [118] generalizing it to manifolds that do not have a metric. The point of view expounded in the books of Weyl and Pauli is that parallel transport is best defined as a mapping of the “totality of all vectors” that “originate” at one point of a manifold with the totality at another point. (In modern texts, one will find discussions based on fiber bundles.) Pauli points out that we cannot simply require equality of vector components as the mapping.
Let us examine the parallel transport of the forcefree, point particle velocity in Euclidean threedimensional space as a means for motivating the form of the mapping. As the velocity is constant, we know that the curve traced out by the particle will be a straight line. In fact, we can turn this around and say that the velocity parallel transports itself because the path traced out is a geodesic (i.e. the straightest possible curve allowed by Euclidean space). In our analysis we will borrow liberally from the excellent discussion of Lovelock and Rund [76]. Their text is comprehensive yet readable for one who is not wellversed with differential geometry. Finally, we note that this analysis will be of use later when we obtain the Newtonian limit of the general relativistic equations, in an arbitrary coordinate basis.
2.3 The Lie derivative and spacetime symmetries
From the above discussion it should be evident that there are other ways to take derivatives in a curved spacetime. A particularly important tool for measuring changes in tensors from point to point in spacetime is the Lie derivative. It requires a vector field, but no connection, and is a more natural definition in the sense that it does not even require a metric. It yields a tensor of the same type and rank as the tensor on which the derivative operated (unlike the covariant derivative, which increases the rank by one). The Lie derivative is as important for Newtonian, nonrelativistic fluids as for relativistic ones (a fact which needs to be continually emphasized as it has not yet permeated the fluid literature for chemists, engineers, and physicists). For instance, the classic papers on the ChandrasekharFriedmanSchutz instability [44, 45] in rotating stars are great illustrations of the use of the Lie derivative in Newtonian physics. We recommend the book by Schutz [104] for a complete discussion and derivation of the Lie derivative and its role in Newtonian fluid dynamics (see also the recent series of papers by Carter and Chamel [22, 23, 24]). We will adapt here the coordinatebased discussion of Schouten [99], as it may be more readily understood by those not wellversed in differential geometry.
In a first course on classical mechanics, when students encounter rotations, they are introduced to the idea of active and passive transformations. An active transformation would be to fix the origin and axisorientations of a given coordinate system with respect to some external observer, and then move an object from one point to another point of the same coordinate system. A passive transformation would be to place an object so that it remains fixed with respect to some external observer, and then induce a rotation of the object with respect to a given coordinate system, by rotating the coordinate system itself with respect to the external observer. We will derive the Lie derivative of a vector by first performing an active transformation and then following this with a passive transformation and finding how the final vector differs from its original form. In the language of differential geometry, we will first “pushforward” the vector, and then subject it to a “pullback”.
As an interesting aside it is worth noting that Arnold [10], only a little whimsically, refers to this construction as the “fisherman’s derivative”. He imagines a fisherman sitting in a boat on a river, “taking derivatives” as the boat moves with the current. Playing on this imagery, the covariant derivative is cast with the hightest Zebco parallel transport fishing pole, the Lie derivative with the Shimano, Liedragging ultralight. Let us now see how Liedragging reels in vectors.
 1.
there exists a Killing vector t^{ μ } that is timelike at spatial infinity;
 2.
there exists a Killing vector ϕ^{ μ } that vanishes on a timelike 2surface (called the axis of symmetry), is spacelike everywhere else, and whose orbits are closed curves; and
 3.
asymptotic flatness means the scalar products t_{ μ }t^{ μ }, and t_{ μ }ϕ^{ μ } tend to, respectively, −1, +∞, and 0 at spatial infinity.
2.4 Spacetime curvature
A more intuitive understanding of the Riemann tensor is obtained by seeing how its presence leads to a pathdependence in the changes that a vector experiences as it moves from point to point in spacetime. Such a situation is known as a “nonintegrability” condition, because the result depends on the whole path and not just the initial and final points. That is, it is unlike a total derivative which can be integrated and thus depends on only the lower and upper limits of the integration. Geometrically we say that the spacetime is curved, which is why the Riemann tensor is also known as the curvature tensor.
To illustrate the meaning of the curvature tensor, let us suppose that we are given a surface that can be parameterized by the two parameters λ and η. Points that live on this surface will have coordinate labels x^{ μ }(λ, η). We want to consider an infinitesimally small “parallelogram” whose four corners (moving counterclockwise with the first corner at the lower left) are given by x^{ μ }(λ, η), x^{ μ }(λ, η + δη), x^{ μ }(λ + δλ, η + δη), and x^{ μ }(λ + δλ, η). Generally speaking, any “movement” towards the right of the parallelogram is effected by varying η, and that towards the top results by varying λ. The plan is to take a vector V^{ μ } (λ, η) at the lowerleft corner x^{ μ }(λ, η), parallel transport it along a λ = const curve to the lowerright corner at x^{ μ }(λ, η + δη) where it will have the components V^{ μ }(λ, η + δη), and end up by parallel transporting V^{ μ } at x^{ μ }(λ, η + δη) along an η = const curve to the upperright corner at x^{ μ }(λ, δλ, η + δη). We will call this path I and denote the final component values of the vector as \(V_{\rm{I}}^\mu\). We repeat the same process except that the path will go from the lowerleft to the upperleft and then on to the upperright corner. We will call this path II and denote the final component values as \(V_{{\rm{II}}}^\mu\).
3 The StressEnergyMomentum Tensor and the Einstein Equations
Without an a priori, physicallybased specification for T_{ μν }, solutions to the Einstein equations are devoid of physical content, a point which has been emphasized, for instance, by Geroch and Horowitz (in [52]). Unfortunately, the following algorithm for producing “solutions” has been much abused: (i) specify the form of the metric, typically by imposing some type of symmetry, or symmetries, (ii) work out the components of G_{ μν } based on this metric, (iii) define the energy density to be G_{00} and the pressure to be G_{11}, say, and thereby “solve” those two equations, and (iv) based on the “solutions” for the energy density and pressure solve the remaining Einstein equations. The problem is that this algorithm is little more than a mathematical game. It is only by sheer luck that it will generate a physically viable solution for a nonvacuum spacetime. As such, the strategy is antithetical to the raison d’être of gravitationalwave astrophysics, which is to use gravitationalwave data as a probe of all the wonderful microphysics, say, in the cores of neutron stars. Much effort is currently going into taking given microphysics and combining it with the Einstein equations to model gravitationalwave emission from realistic neutron stars. To achieve this aim, we need an appreciation of the stressenergy tensor and how it is obtained from microphysics.
Those who are familiar with Newtonian fluids will be aware of the roles that total internal energy, particle flux, and the stress tensor play in the fluid equations. In special relativity we learn that in order to have spacetime covariant theories (e.g. wellbehaved with respect to the Lorentz transformations) energy and momentum must be combined into a spacetime vector, whose zeroth component is the energy and the spatial components give the momentum. The fluid stress must also be incorporated into a spacetime object, hence the necessity for T_{ μν }. Because the Einstein tensor’s covariant divergence vanishes identically, we must have also \({\nabla _\mu}{T^\mu}_\nu = 0\) (which we will later see happens automatically once the fluid field equations are satisfied).
4 Why Are Fluids Useful Models?
The MerriamWebster online dictionary (http://www.mw.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 socalled “force” carriers (gaugevector 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.
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 threedimensional 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 socalled threebrane (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 threedimensional “matter” space (i.e. the lefthandside of Figure 7).
Once one understands how to build a fluid model using the matter space, it is straightforward 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 nonzero temperature, i.e. nonzero 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 fourvelocity 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 socalled 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 He^{4} [94] at nonzero temperature and a mixture of superfluid He^{4} and He^{3} [8].
5 A Primer on Thermodynamics and Equations of State
Fluids consist of many fluid elements, and each fluid element consists of many particles. The state of matter in a given fluid element is determined thermodynamically [96], meaning that only a few parameters are tracked as the fluid element evolves. Generally, not all the thermodynamic variables are independent, being connected through the socalled equation of state. The number of independent variables can also be reduced if the system has an overall additivity property. As this is a very instructive example, we will now illustrate such additivity in detail.
5.1 Fundamental, or Euler, relation
We can think of a given relation ρ(s, n) as the equation of state, to be determined in the flat, tangent space at each point of the manifold, or, physically, small enough patches across which the changes in the gravitational field are negligible, but also large enough to contain a large number of particles. For example, for a neutron star Glendenning [48] has reasoned that the relative change in the metric over the size of a nucleon with respect to the change over the entire star is about 10^{−19}, and thus one must consider many internucleon spacings before a substantial change in the metric occurs. In other words, it is sufficient to determine the properties of matter in special relativity, neglecting effects due to spacetime curvature. The equation of state is the major link between the microphysics that governs the local fluid behavior and global quantities (such as the mass and radius of a star).
5.2 From microscopic models to the fluid equation of state
Let us now briefly discuss how an equation of state is constructed. For simplicity, we focus on a oneparameter system, with that parameter being the particle number density. The equation of state will then be of the form ρ = ρ(n). In manybody physics (such as studied in condensed matter, nuclear, and particle physics) one can in principle construct the quantum mechanical particle number density n_{QM}, stressenergymomentum tensor \(T_{\mu \nu}^{QM}\), and associated conserved particle number density current \(n_{QM}^\mu\) (starting with some fundamental Lagrangian, say; cf. [115, 48, 116]). But unlike in quantum field theory in curved spacetime [13], one assumes that the matter exists in an infinite Minkowski spacetime (cf. the discussion following Equation (85)). If the reader likes, the application of \(T_{\mu \nu}^{QM}\) at a spacetime point means that \(T_{\mu \nu}^{QM}\) has been determined with respect to a flat tangent space at that point.
One must be very careful to distinguish \(T_{\mu \nu}^{QM}\) from T_{ μν }. The former describes the states of elementary particles with respect to a fluid element, whereas the latter describes the states of fluid elements with respect to the system. Comer and Joynt [35] have shown how this line of reasoning applies to the twofluid case.
6 An Overview of the Perfect Fluid
There are many different ways of constructing general relativistic fluid equations. Our purpose here is not to review all possible methods, but rather to focus on a couple: (i) an “offtheshelve” consistency analysis for the simplest fluid a la Eckart [39], to establish some key ideas, and then (ii) a more powerful method based on an action principle that varies fluid element world lines. The ideas behind this variational approach can be traced back to Taub [108] (see also [101]). Our description of the method relies heavily on the work of Brandon Carter, his students, and collaborators [19, 36, 37, 28, 29, 67, 90, 91]. We prefer this approach as it utilizes as much as possible the tools of the trade of relativistic fields, i.e. no special tricks or devices will be required (unlike even in the case of our “offtheshelve” approach). One’s footing is then always made sure by wellgrounded, actionbased derivations. As Carter has always made clear: When there are multiple fluids, of both the charged and uncharged variety, it is essential to distinguish the fluid momenta from the velocities, in particular in order to make the geometrical and physical content of the equations transparent. A wellposed action is, of course, perfect for systematically constructing the momenta.
6.1 Ratesofchange and Eulerian versus Lagrangian observers
The key geometric difference between generally covariant Newtonian fluids and their general relativistic counterparts is that the former have an a priori notion of time [22, 23, 24]. Newtonian fluids also have an a priori notion of space (which can be seen explicitly in the Newtonian covariant derivative introduced earlier; cf. the discussion in [22]). Such a structure has clear advantages for evolution problems, where one needs to be unambiguous about the rateofchange of the system. But once a problem requires, say, electromagnetism, then the a priori Newtonian time is at odds with the full spacetime covariance of electromagnetic fields. Fortunately, for spacetime covariant theories there is the socalled “3 + 1” formalism (see, for instance, [105]) that allows one to define “ratesofchange” in an unambiguous manner, by introducing a family of spacelike hypersurfaces (the “3”) given as the level surfaces of a spacetime scalar (the “1”).
Something that Newtonian and relativistic fluids have in common is that there are preferred frames for measuring changes — those that are attached to the fluid elements. In the parlance of hydrodynamics, one refers to Lagrangian and Eulerian frames, or observers. A Newtonian Eulerian observer is one who sits at a fixed point in space, and watches fluid elements pass by, all the while taking measurements of their densities, velocities, etc. at the given location. In contrast, a Lagrangian observer rides along with a particular fluid element and records changes of that element as it moves through space and time. A relativistic Lagrangian observer is the same, but the relativistic Eulerian observer is more complicated to define. Smarr and York [105] define such an observer as one who would follow along a worldline that remains everywhere orthogonal to the family of spacelike hypersurfaces.
The existence of a preferred frame for a one fluid system can be used to great advantage. In the next Section 6.2 we will use an “offtheshelf” analysis that exploits a preferred frame to derive the standard perfect fluid equations. Later, we will use Eulerian and Lagrangian variations to build an action principle for the single and multiple fluid systems. These same variations can also be used as the foundation for a linearized perturbation analysis of neutron stars [63]. As we will see, the use of Lagrangian variations is absolutely essential for establishing instabilities in rotating fluids [44, 45]. Finally, we point out that multiple fluid systems can have as many notions of Lagrangian observers as there are fluids in the system.
6.2 The single, perfect fluid problem: “Offtheshelf” consistency analysis
We earlier took the components of a general stressenergymomentum tensor and projected them onto the axes of a coordinate system carried by an observer moving with fourvelocity U^{ μ }. As mentioned above, the simplest fluid is one for which there is only one fourvelocity u^{ μ }. Hence, there is a preferred frame defined by u^{ μ }, and if we want the observer to sit in this frame we can simply take U^{ μ } = u^{ μ }. With respect to the fluid there will be no momentum flux, i.e. \({{\mathcal P}_\mu} = 0\). Since we use a fully spacetime covariant formulation, i.e. there are only spacetime indices, the resulting stressenergymomentum tensor will transform properly under general coordinate transformations, and hence can be used for any observer.
Given that we have just inferred the equations of motion from the identity that \({\nabla _\mu}{T^\mu}_\nu = 0\), we now emphatically state that while the equations are correct the reasoning is severely limited. In fact, from a field theory point of view it is completely wrong! The proper way to think about the identity is that the equations of motion are satisfied first, which then guarantees that \({\nabla _\mu}{T^\mu}_\nu = 0\). There is no clearer way to understand this than to study the multifluid case: Then the vanishing of the covariant divergence represents only four equations, whereas the multifluid problem clearly requires more information (as there are more velocities that must be determined). We have reached the end of the road as far as the “offtheshelf” strategy is concerned, and now move on to an actionbased derivation of the fluid equations of motion.
7 Setting the Context: The Point Particle
The simplest physics problem, i.e. the point particle, has always served as a guide to deep principles that are used in much harder problems. We have used it already to motivate parallel transport as the foundation for the covariant derivative. Let us call upon the point particle again to set the context for the actionbased derivation of the fluid field equations. We will simplify the discussion by considering only motion in one dimension. We assure the reader that we have good reasons, and ask for patience while we remind him/her of what may be very basic facts.
8 The “Pullback” Formalism for a Single Fluid
In this section the equations of motion and the stressenergymomentum tensor for a onecomponent, general relativistic fluid are obtained from an action principle. Specifically a socalled “pullback” approach (see, for instance, [36, 37, 34]) is used to construct a Lagrangian displacement of the number density fourcurrent n^{ μ }, whose magnitude n is the particle number density. This will form the basis for the variations of the fundamental fluid variables in the action principle.
The main reason for introducing the dual is that it is straightforward to construct a particle number density threeform that is automatically closed, since the conservation of the particle number density current should not — speaking from a strict field theory point of view — be a part of the equations of motion, but rather should be automatically satisfied when evaluated on a solution of the “true” equations.
We should point out that our consideration of a master function of the form Λ = Λ(n^{2}) is based, in part, on the assumptions that the matter is locally isotropic, meaning that there are no locally preferred directions (such as in a neutron star crust) or another covector A^{ μ } to form the additional scalar n^{ μ }A_{ μ } (as would be the case with coupling to electromagnetism, say). A term that could be added is of the form n^{ μ }∇_{ μ }ϕ for an arbitrary scalar field ϕ. Unlike the previous two possible additional terms, it would not affect the equations of motion, since ∇_{ μ }n^{ μ } = 0 by construction, and an integration by parts generates a boundary term. Our point of view is that the master function is fixed by the local microphysics of the matter; (cf. the discussion in Section 5.2).
Let us now recall the discussion of the point particle. There we pointed out that only the fully conservative form of Newton’s Second Law follows from an action, i.e. external or dissipative forces are excluded. However, we argued that a wellestablished form of Newton’s Second Law is known that allows for external and/or dissipative forces (cf. Equation (120)). There is thus much purpose in using the particular symbol f_{ ν } in Equation (152). We may take the f_{ ν } to be the relativistic analogue of the lefthandside of Equation (120) in every sense. In particular, when dissipation and/or external “forces” act in a general relativistic setting, they are incorporated via the righthandside of Equation (152).
9 The TwoConstituent, Single Fluid
Given that we only have one fourvelocity, the system will still just have one fluid element per spacetime point. But unlike before, there will be an additional conserved number, N_{s}, that can be attached to each worldline, like the particle number N_{n} of Figure 7. In order to describe the worldlines we can use the same three scalars X^{ A }(x^{ μ }) as before. But how do we get a construction that allows for the additional conserved number? Recall that the intersections of the worldlines with some hypersurface, say t = 0, is uniquely specified by the three X^{ A }(0, x^{ i }) scalars. Each worldline will have also the conserved numbers N_{n} and N_{s} assigned to them. Thus, the values of these numbers can be expressed as functions of the X^{ A }(0, x^{ i }). But most importantly, the fact that each N_{x} is conserved, means that this function specification must hold for all of spacetime, so that in particular the ratio x_{s} is of the form x_{s}(x^{ μ }) = x_{s}(X^{ A }(x^{ μ })). Consequently, we now have a construction whereby this ratio identically satisfies Equation (154), and the action principle remains a variational problem just in terms of the three X^{ A } scalars.
10 The “PullBack” Formalism for Two Fluids
Having discussed the single fluid model, and how one accounts for stratification, it is time to move on to the problem of modeling multifluid systems. We will experience for the first time novel effects due to the presence of a relative flow between two interpenetrating fluids, and the fact that there is no longer a single, preferred restframe. This kind of formalism is necessary, for example, for the simplest model of a neutron star, since it is generally accepted that the inner crust is permeated by an independent neutron superfluid, and the outer core is thought to contain superfluid neutrons, superconducting protons, and a highly degenerate gas of electrons. Still unknown is the number of independent fluids that would be required for neutron stars that have quark matter in the deep core [1]. The model can also be used to describe superfluid Helium and heatconducting fluids, which is of importance for incorporation of dissipation (see Section 14). We will focus on this example here. It should be noted that, even though the particular system we concentrate on consists of only two fluids, it illustrates all new features of a general multifluid system. Conceptually, the greatest step is to go from one to two fluids. A generalization to a system with more degrees of freedom is straightforward.
In the general case, the momentum of one constituent carries along some mass current of the other constituents. The entrainment only vanishes in the special case where Λ is independent of \(n_{{\rm{xy}}}^2({\rm{x}} \neq {\rm{y}})\) because then we obviously have \({{\mathcal A}^{{\rm{xy}}}} = 0\). Entrainment is an observable effect in laboratory superfluids [94, 110] (e.g. via flow modifications in superfluid ^{4}He and mixtures of superfluid ^{3}He and ^{4}He). In the case of neutron stars, entrainment is an essential ingredient of the current best explanation for the socalled glitches [95, 97]. Carter [19] has also argued that these “anomalous” terms are necessary for causally wellbehaved heat conduction in relativistic fluids, and by extension necessary for building wellbehaved relativistic equations that incorporate dissipation.
It must be noted that Equation (183) is significantly different from the multiconstituent version of Equation (166). This is true even if one is solving for a static and spherically symmetric configuration, where the fluid fourvelocities would all necessarily be parallel. Simply put, Equation (183) still represents two independent equations. If one takes entropy as an independent fluid, then the static and spherically symmetric solutions will exhibit thermal equilibrium [38]. This explains, for instance, why one must specify an extra condition (e.g. convective stability [117]) to solve for a doubleconstituent star with only one fourvelocity.
11 Speeds of Sound
Crucial to the understanding of black holes is the effect of spacetime curvature on the lightcone structures, that is, the totality of null vectors that emanate from each spacetime point. Crucial to the propagation of massless fields is the lightcone structure. In the case of fluids, it is both the speed of light and the speed (and/or speeds) of sound that dictate how waves propagate through the matter, and thus how the matter itself propagates. We give here a simple analysis that uses planewave propagation to derive the speed(s) of sound in the singlefluid, twoconstituent singlefluid, and twofluid cases, assuming that the metric variations vanish (see [19] for a more rigorous derivation). The analysis is local, assuming that the speed of sound is a locally defined quantity, and performed using local, Minkowski coordinates x^{ μ }. The purpose of the analysis is to illuminate how the presence of various constituents and multifluids impact the local dynamics.
11.1 Single fluid case
11.2 Twoconstituent, single fluid case
11.3 Two fluid case
The twofluid problem is qualitatively different from the previous two cases, since there are now two independent density currents. This fact impacts the analysis in two crucial ways: (i) The master function Λ depends on \(n_{\rm{n}}^2\), \(n_{\rm{s}}^2\), and \(n_{{\rm{ns}}}^2 = n_{{\rm{sn}}}^2\) (i.e. entrainment is present), and (ii) the equations of motion, after taking into account the transverse flow condition, are doubled to \(\delta f_\mu ^{\rm{n}} = 0 = \delta f_\mu ^{\rm{s}}\). As we will see, the key point is that there are now two sound speeds that must be determined.
The sound speed analysis is local, but its results are seen globally in the analysis of modes of oscillation of a fluid body. For a neutron star, the full spectrum of modes is quite impressive (see McDermott et al. [77]): polar (or spheroidal) f, p, and gmodes, and the axial (or toroidal) rmodes. Epstein [41] was the first to suggest that there should be even more modes in superfluid neutron stars because the superfluidity allows the neutrons to move independently of the protons. Mendell [78] developed this idea further by using an analogy with coupled pendulums. He argued that the new modes should feature a countermotion between the neutrons and protons, i.e. as the neutrons move out radially, say, the protons will move in. This is in contrast to ordinary fluid motion that would have the neutrons and protons move in more or less “lockstep”. Analytical and numerical studies [70, 74, 38, 5] have confirmed this basic picture and the new modes of oscillation are known as superfluid modes.
12 The Newtonian Limit and the Euler Equations
One reason relativistic fluids are needed is that they can be used to model neutron stars. However, even though neutron stars are clearly general relativistic objects, one often starts with good old Newtonian physics when considering new applications. The main reason is that effects (such as acoustic modes of oscillation) which are primarily due to the fluids that make up the star can often be understood qualitatively from Newtonian calculations. There are also certain regimes in a neutron star where the Newtonian limit is sufficient quantitatively (such as the outer layers).
There has been much progress recently in the analysis of Newtonian multiple fluid systems. Prix [91] has developed an actionbased formalism, analogous to what has been used here. Carter and Chamel [22, 23, 24] have done the same except that they use a fully spacetime covariant formalism. We will be somewhat less ambitious (for example, as in [5]) by extracting the Newtonian equations as the nonrelativistic limit of the fully relativistic equations. Given the results of Prix as well as of Carter and Chamel we can think of this exercise as a consistency check of our equations.
A Newtonian twofluid system can be obtained in a similar fashion. As discussed in Section 10, the main difference is that we need two sets of worldlines, describable, say, by curves \(x_{\rm{x}}^\mu ({\tau _{\rm{x}}})\) where τ_{x} is the proper time along a constituent’s worldline. Of course, entrainment also comes into play. Its presence implies that the relative flow of the fluids is required to specify the local thermodynamic state of the system, and that the momentum of a given fluid is not simply proportional to that fluid’s flux. This is the situation for superfluid He^{4} [94, 110], where the entropy can flow independently of the superfluid Helium atoms. Superfluid He^{3} can also be included in the mixture, in which case there will be a relative flow of the He^{3} isotope with respect to He^{4}, and relative flows of each with respect to the entropy [113].
13 The CFS Instability
Investigations into the stability properties of rotating selfgravitating bodies are of obvious relevance to astrophysics. By improving our understanding of the relevant issues we can hope to shed light on the nature of the various dynamical and secular instabilities that may govern the spinevolution of rotating stars. The relevance of such knowledge for neutron star astrophysics may be highly significant, especially since instabilities may lead to detectable gravitationalwave signals. In this section we will review the Lagrangian perturbation framework developed by Friedman and Schutz [44, 45] for rotating nonrelativistic stars. This will lead to criteria that can be used to decide when the oscillations of a rotating neutron star are unstable. We provide an explicit example proving the instability of the socalled rmodes at all rotation rates in a perfect fluid star.
13.1 Lagrangian perturbation theory
The trivials have the potential to cause trouble because they affect the canonical energy. Before one can use the canonical energy to assess the stability of a rotating configuration one must deal with this “gauge problem”. To do this one should ensure that the displacement vector ξ is orthogonal to all trivials. A prescription for this is provided by Friedman and Schutz [44]. In particular, they show that the required canonical perturbations preserve the vorticity of the individual fluid elements. Most importantly, one can also prove that a normal mode solution is orthogonal to the trivials. Thus, normal mode solutions can serve as canonical initial data, and be used to assess stability.
13.2 Instabilities of rotating perfect fluid stars

Dynamical instabilities are only possible for motions such that E_{c} = 0. This makes intuitive sense since the amplitude of a mode for which E_{c} vanishes can grow without bounds and still obey the conservation laws.

If the system is coupled to radiation (e.g. gravitational waves) which carries positive energy away from the system (which should be taken to mean that ∂_{t}E_{ c } < 0) then any initial data for which E_{ c } < 0 will lead to an unstable evolution.
Equation (256) forms a key part of the proof that rotating perfect fluid stars are generically unstable in the presence of radiation [45]. The argument goes as follows: Consider modes with finite frequency in the Ω → 0 limit. Then Equation (256) implies that corotating modes (with σ_{p} > 0) must have J_{c} > 0, while counterrotating modes (for which σ_{p} < 0) will have J_{c} < 0. In both cases E_{c} > 0, which means that both classes of modes are stable. Now consider a small region near a point where σ_{p} = 0 (at a finite rotation rate). Typically, this corresponds to a point where the initially counterrotating mode becomes corotating. In this region J_{c} < 0. However, E_{c} will change sign at the point where σ_{p} (or, equivalently, the frequency ω) vanishes. Since the mode was stable in the nonrotating limit this change of sign indicates the onset of instability at a critical rate of rotation.
13.3 The rmode instability
13.4 The relativistic problem
The theoretical framework for studying stellar stability in general relativity was mainly developed during the 1970s, with key contributions from Chandrasekhar and Friedman [31, 32] and Schutz [102, 103]. Their work extends the Newtonian analysis discussed above. There are basically two reasons why a relativistic analysis is more complicated than the Newtonian one. Firstly, the problem is algebraically more complex because one must solve the Einstein field equations in addition to the fluid equations of motion. Secondly, one must account for the fact that a general perturbation will generate gravitational waves. The work culminated in a series of papers [43, 44, 45, 42] in which the role that gravitational radiation plays in these problems was explained, and a foundation for subsequent research in this area was established. The main result was that gravitational radiation acts in the same way in the full theory as in a postNewtonian analysis of the problem. If we consider a sequence of equilibrium models, then a mode becomes secularly unstable at the point where its frequency vanishes (in the inertial frame). Most importantly, the proof does not require the completeness of the modes of the system.
14 Modelling Dissipation
Although the inviscid model provides a natural starting point for any investigation of the dynamics of a fluid system, the effects of dissipative mechanisms are often key to the construction of a realistic model. Consider, for example, the case of neutron star oscillations and possible instabilities. While it is interesting from the conceptual point of view to establish that an instability (such as the gravitationalwave driven instability of the fundamental fmode or the inertial rmode discussed above) may be present in an ideal fluid, it is crucial to establish that the instability actually has opportunity to grow on a reasonably short timescale. To establish this, one must consider the most important damping mechanisms and work out whether they will suppress the instability or not. A recent discussion of these issues in the context of the rmode instability can be found in [4].
From the point of view of relativistic fluid dynamics, it is clear already from the outset that we are facing a difficult problem. After all, the Fourier theory of heat conduction leads to instantaneous propagation of thermal signals. The fact that this noncausality is built into the description is unattractive already in the context of the NavierStokes equations. After all, one would expect heat to propagate at roughly the mean molecular speed in the system. For a relativistic description noncausal behavior would be truly unacceptable. As work by Lindblom and Hiscock [53] has established, there is a deep connection between causality, stability, and hyperbolicity of a dissipative model. One would expect an acceptable model to be hyperbolic, not allowing signals to propagate superluminally.
Our aim in this section is to discuss the three main models that exist in the literature. We first consider the classic work of Eckart [39] and Landau and Lifshitz [66], which is based on a seemingly natural extension of the inviscid equations. However, a detailed analysis of Lindblom and Hiscock [54, 55] has demonstrated that these descriptions have serious flaws and must be considered unsuitable for practical use. However, having discussed these models it is relatively easy to extend them in the way proposed by Israel and Stewart [107, 57, 58]. Their description, the derivation of which was inspired by early work of Grad [50] and Müller [81] and results from relativistic kinetic theory, provides a framework that is generally accepted as meeting the key criteria for a relativistic model [53]. Finally, we describe Carter’s more recent approach to the problem. This model is elegant because it makes maximal use of a variational argument. The construction is also more general than that of Israel and Stewart. In particular, it shows how one would account for several dynamically independent interpenetrating fluid species. This extension is important for, for example, the consideration of relativistic superfluid systems.
14.1 The “standard” relativistic models
14.2 The IsraelStewart approach
From the above discussion we know that the most obvious strategy for extending relativistic hydrodynamics to include dissipative processes leads to unsatisfactory results. At first sight, this may seem a little bit puzzling because the approach we took is fairly general. Yet, the formulation suffers from pathologies. Most importantly, we have not managed to enforce causality. Let us now explain how this problem can be solved.
Although the IsraelStewart model resolves the problems of the firstorder descriptions for near equilibrium situations, difficult issues remain to be understood for nonlinear problems. This is highlighted in work by Hiscock and Lindblom [56], and Olson and Hiscock [85]. They consider nonlinear heat conduction problems and show that the IsraelStewart formulation becomes noncausal and unstable for sufficiently large deviations from equilibrium. The problem appears to be more severe in the Eckart frame [56] than in the frame advocated by Landau and Lifshitz [85]. The fact that the formulation breaks down in nonlinear problems is not too surprising. After all, the basic foundation is a “Taylor expansion” in the various fields. However, it raises important questions. There are many physical situations where a reliable nonlinear model would be crucial, e.g. heavyion collisions and supernova core collapse. This problem requires further thought.
14.3 Carter’s canonical framework
The most recent attempt to construct a relativistic formalism for dissipative fluids is due to Carter [20]. His approach is less “phenomenological” than the ones we have considered so far in that it is based on making maximal use of variational principle arguments. The construction is also extremely general. On the one hand this makes it more complex. On the other hand this generality could prove useful in more complicated cases, e.g. for investigations of multifluid dynamics and/or elastic media. Given the potential that this formalism has for future applications, it is worth discussing it in detail.
The overall aim is to generalize the variational formulation described in Section 8 in such a way that viscous “stresses” are accounted for. Because the variational foundations are the same, the number currents \(n_{\rm{x}}^\mu\) play a central role (x is a constituent index as before). In addition we can introduce a number of viscosity tensors \(\mathop \tau \nolimits_\sum ^{\mu \nu } \), which we assume to be symmetric (even though it is clear that such an assumption is not generally correct [6]). The index Σ is “analogous” to the constituent index, and represents different viscosity contributions. It is introduced in recognition of the fact that it may be advantageous to consider different kinds of viscosity, e.g. bulk and shear viscosity, separately. As in the case of the constituent index, a repeated index Σ does not imply summation.
A detailed comparison between Carter’s formalism and the IsraelStewart framework has been carried out by Priou [89]. He concludes that the two models, which are both members of a large family of dissipative models, have essentially the same degree of generality and that they are equivalent in the limit of linear perturbations away from a thermal equilibrium state. Providing explicit relations between the main parameters in the two descriptions, he also emphasizes the key point that analogous parameters may not have the same physical interpretation.
In developing his theoretical framework, Carter argued in favor of an “off the peg” model for heat conducting models [18]. This model is similar to that introduced in Section 10, and was intended as a simple, easier to use alternative to the IsraelStewart construction. In the particular example discussed by Carter, he chooses to set the entrainment between particles and entropy to zero. This was done in order to simplify the discussion. But, as a discussion by Olson and Hiscock [87] shows, it has disastrous consequences. The resulting model violates causality in two simple model systems. As discussed by Priou [89] and Carter and Khalatnikov [25], this breakdown emphasizes the importance of the entrainment effect in these problems.
14.4 Remaining issues
We have discussed some of the models that have been constructed in order to incorporate dissipative effects in a relativistic fluid description. We have seen that the most obvious ways of doing this, the “textbook” approach of Eckart [39] and Landau and Lifshitz [66], fail completely. They do not respect causality and have serious stability problems. We have also described how this problem can be fixed by introducing additional dynamical fields. We discussed the formulations of Israel and Stewart [107, 57, 58] and Carter [20] in detail. From our discussion it should be clear that these models are examples of an extremely large family of possible theories for dissipative relativistic fluids. Given this wealth of possibilities, can we hope to find the “right” model? To some extent, the answer to this question relies on the extra parameters one has introduced in the theory. Can they be constrained by observations? This question has been discussed by Geroch [47] and Lindblom [73]. The answer seems to be no, we should not expect to be able to use observations to single out a preferred theoretical description. The reason for this is that the different models relax to the NavierStokes form on very short timescales. Hence, one will likely only be able to constrain the standard shear and bulk viscosity coefficients, etc. Related questions concern the practicality of the different proposed schemes. To a certain extent, this is probably a matter of taste. Of course, it may well be that the additional parameters required in a particular model are easier to extract from microphysics arguments. This could make this description easier to use in practice, which would be strong motivation for preferring it. Of course, there is no guarantee that the same formulation will be ideal for all circumstances. Clearly, there is scope for a lot more research in this problem area.
15 Heavy Ion Collisions
Relativistic fluid dynamics has regularly been used as a tool to model heavy ion collisions. The idea of using hydrodynamics to study the process of multiparticle production in highenergy hadron collisions can be traced back to work by, in particular, Landau in the early 1950s (see [12]). In the early days these phenomena were observed in cosmic rays. The idea to use hydrodynamics was resurrected as collider data became available [15] and early simulations were carried out at Los Alamos [2, 3]. More recently, modeling has primarily been focussed on reproducing data from, for example, CERN. A useful review of this active area of research can be found in [33].
In a hydrodynamics based model, a highenergy nuclear collision is viewed in the following way: In the centerofmass frame two Lorentz contracted nuclei collide and, after a complex microscopic process, a hot dense plasma is formed. In the simplest description this matter is assumed to be in local thermal equilibrium. The initial thermalization phase is, of course, out of reach for hydrodynamics. In the model, the state of matter is simply specified by the initial conditions, e.g. in terms of distributions of fluid velocities and thermodynamical quantities. Then follows a hydrodynamical expansion, which is described by the standard conservation equations for energy/momentum, baryon number, and other conserved quantities, such as strangeness, isotopic spin, etc. (see [40] for a variational principle derivation of these equations). As the expansion proceeds, the fluid cools and becomes increasingly rarefied. This eventually leads to the decoupling of the constituent particles, which then do not interact until they reach the detector.
Fluid dynamics provides a well defined framework for studying the stages during which matter becomes highly excited and compressed and, later, expands and cools down. In the final stage when the nuclear matter is so dilute that nucleonnucleon collisions are infrequent, hydrodynamics ceases to be valid. At this point additional assumptions are necessary to predict the number of particles, and their energies, which may be formed (to be compared to data obtained from the detector). These are often referred to as the “freezeout” conditions. The problem is complicated by the fact that the “freezeout” typically occurs at a different time for each fluid cell.
 1.
many degrees of freedom in the system,
 2.
a short mean free path,
 3.
a short mean stopping length,
 4.
a sufficient reaction time for thermal equilibration, and
 5.
a short de Broglie wavelength (so that quantum mechanics can be ignored).
Another key reason why hydrodynamic models are favored is the simplicity of the input. Apart from the initial conditions which specify the masses and velocities, one needs only an equation of state and an Ansatz for the thermal degrees of freedom. If one includes dissipation one must in addition specify the form and magnitude of the viscosity and heat conduction. The fundamental conservation laws are incorporated into the Euler equations. In return for this relatively modest amount of input, one obtains the differential cross sections of all the final particles, the composition of clusters, etc. Of course, before one can confront the experimental data, one must make additional assumptions about the freezeout, chemistry, etc. A clear disadvantage of the hydrodynamics model is that much of the microscopic dynamics is lost.
Let us discuss some specific aspects of the hydrodynamics that has been used in this area. As we will recognize, the issues that need to be addressed for heavyion collisions are very similar to those faced in studies of relativistic dissipation theory and multifluid modeling. The one key difference is that the problem only requires special relativity, so there is no need to worry about the spacetime geometry. Of course, it is still convenient to use a fully covariant description since one is then not tied down to the use of a particular set of coordinates.
In many studies of heavy ions a particular frame of reference is chosen. As we know from our discussion of dissipation and causality (see Section 14), this is an issue that must be approached with some care. In the context of heavyion collisions it is common to choose u^{ μ } as the velocity of either energy transport (the LandauLifshitz frame) or particle transport (the Eckart frame). It is recognized that the Eckart formulation is somewhat easier to use and that one can let u^{ μ } be either the velocity of nucleon or baryon number transport. On the other hand, there are cases where the LandauLifshitz picture has been viewed as more appropriate. For instance, when ultrarelativistic nuclei collide they virtually pass through one another leaving the vacuum between them in a highly excited state causing the creation of numerous particleantiparticle pairs. Since the net baryon number in this region vanishes, the Eckart definition of the fourvelocity cannot easily be employed. This discussion is a clear reminder of the situation for viscosity in relativity, and the resolution is likely the same. A true frameindependent description will need to include several distinct fluid components.
Multifluid models have, in fact, often been considered for heavyion collisions. One can, for example, treat the target and projectile nuclei as separate fluids to admit interpenetration, thus arriving at a twofluid model. One could also use a relativistic multifluid model to allow for different species, e.g. nucleons, deltas, hyperons, pions, kaons, etc. Such a model could account for the varying dynamics of the different species and their mutual diffusion and chemical reactions. The derivation of such a model would follow closely our discussion in Section 10. In the heavyion community, it has been common to confuse the issue somewhat by insisting on choosing a particular local rest frame at each spacetime point. This is, of course, complicated since the different fluids move at different speeds relative to any given frame. For the purpose of studying heavy ion collisions in the baryonrich regions of space, the standard option seems to be to define the “baryonic Lorentz frame”. This is the local Lorentz frame in which the motion of the centerofbaryon number (analogous to the centerofmass) vanishes.
The main problem with the onefluid hydrodynamics model is the requirement of thermal equilibrium. In the hydrodynamic equations of motion it is implicitly assumed that local thermal equilibrium is “imposed” via an equation of state of the matter. This means that the relaxation timescale and the meanfree path should be much smaller than both the hydrodynamical timescale and the spatial size of the system. It seems reasonable to wonder if these conditions can be met for hadronic and nuclear collisions. On the other hand, from the kinematical point of view, apart from the use of the equation of state, the equations of hydrodynamics are nothing but conservation laws of energy and momentum, together with other conserved quantities such as charge. In this sense, for any process where the dynamics of flow is an important factor, a hydrodynamic framework is a natural first step. The effects of a finite relaxation time and meanfree path might be implemented at a later stage by using an effective equation of state, incorporating viscosity and heat conductivity, or some simplified transport equations. This does, of course, lead us back to the challenging problem of designing a causal relativistic theory for dissipation (see Section 14). In the context of heavyion collisions no calculations have yet been performed using a fully threedimensional, relativistic theory which includes dissipation. In fact, considering the obvious importance of entropy, it is surprising that so few calculations have been reported for either relativistic or nonrelativistic hydrodynamics (although see [59]). An interesting comparison of different dissipative formulations is provided in [82, 83].
16 Superfluids and Broken Symmetries
In this section we discuss models that result when additional constraints are made on the properties of the fluids. We focus on the modeling of superfluid systems, using as our example the case of superfluid He^{4} [61, 94, 110]. The equations describing more complex systems are readily obtained by generalizing our discussion. We contrast three different descriptions: (i) the model that follows from the variational framework that has been our prime focus so far if we impose that one constituent is irrotational, (ii) the potential formulation due to Khalatnikov and collaborators, and (iii) a “hybrid” formulation which has been used in studies of heavyion collisions with broken symmetries.
16.1 Superfluids
Neutron star physics provides ample motivation for the need to develop a relativistic description of superfluid systems. As the typical core temperatures (below 10^{8} K) are far below the Fermi temperature of the various constituents (of the order of 10^{12} K for baryons) neutron stars are extremely cold on the nuclear temperature scale. This means that, just like ordinary matter at near absolute zero temperature, the matter in the star will most likely freeze to a solid or become superfluid. While the outer parts of the star, the socalled crust, form an elastic lattice, the inner parts of the star are expected to be superfluid. In practice, this means that we must be able to model mixtures of superfluid neutrons and superconducting protons. It is also likely that we need to understand superfluid hyperons and color superconducting quarks. There are many hard physics questions that need to be considered if we are to make progress in this area. In particular, we need to make contact with microphysics calculations that determine the various parameters of such multifluid systems. However, we will ignore this aspect and focus on the various fluid models that have been used to describe relativistic superfluids.
One of the key features of a pure superfluid is that it is irrotational. Bulk rotation is mimicked by the formation of vortices, slim “tornadoes” representing regions where the superfluid degeneracy is broken. In practice, this means that one would often, e.g. when modeling global neutron star oscillations, consider a macroscopic model where one “averages” over a large number of vortices. The resulting model would closely resemble the standard fluid model. Of course, it is important to remember that the vortices are present on the microscopic scale, and that they may affect the values of various parameters in the problem. There are also unique effects that are due to the vortices, e.g. the mutual friction that is thought to be the key agent that counteracts relative rotation between the neutrons and protons in a superfluid neutron star core [79].
16.2 Broken symmetries
In the context of heavyion collisions, models accounting for broken symmetries have sometimes been considered. At a very basic level, a model with a broken U(1) symmetry should correspond to the superfluid model described above. However, at first sight our equations differ from those used, for example, in [106, 92, 125]. Since we are keen to convince the reader that the variational framework we have discussed in this article is able to cover all cases of interest (in fact, we believe that it is more powerful than alternative formulations) a demonstration that we can reformulate our equations to get those written down for a system with a broken U(1) symmetry has some merit. The exercise is also of interest since it connects with models that have been used to describe other superfluid systems.
17 Final Remarks
In writing this review, we have tried to discuss the different building blocks that are needed if one wants to construct a relativistic theory for fluids. Although there are numerous alternatives, we opted to base our discussion of the fluid equations of motion on the variational approach pioneered by Taub [108] and in recent years developed considerably by Carter [17, 19, 21]. This is an appealing strategy because it leads to a natural formulation for multifluid problems. Having developed the variational framework, we discussed applications. Here we had to decide what to include and what to leave out. Our decisions were not based on any particular logic, we simply included topics that were either familiar to us, or interested us at the time. That may seem a little peculiar, but one should keep in mind that this is a “living” review. Our intention is to add further applications when the article is updated. On the formal side, we could consider how one accounts for elastic media and magnetic fields, as well as technical issues concerning relativistic vortices (and cosmic strings). On the application side, we may discuss many issues for astrophysical fluid flows (like supernova core collapse, jets, gammaray bursts, and cosmology).
In updating this review we will obviously also correct the mistakes that are sure to be found by helpful colleagues. We look forward to receiving any comments on this review. After all, fluids describe physics at many different scales and we clearly have a lot of physics to learn. The only thing that is certain is that we will enjoy the learning process!
Footnotes
 1.
In this article we use “superfluid” to refer to any system which has the ability to flow without friction. In this sense, superfluids and superconductors are viewed in the same way. When we wish to distinguish charge carrying fluids, we will call them superconductors.
 2.
There are three space and one time dimensions that form a type of topological space known as a manifold [114]. This means that local, suitably small patches of a curved spacetime are practically the same as patches of flat, Minkowski spacetime. Moreover, where two patches overlap, the identification of points in one patch with those in the other is smooth.
 3.
We say “combined” here because the First Law is a statement about heat and work, and says nothing about the entropy, which enters through the Second Law. Heat is not strictly equal to T dS for all processes; they are equal for quasistatic processes, but not for free expansion of a gas into vacuum [100].
 4.
It is worth pointing out that we are restricting the problem somewhat by imposing particle conservation already from the outset. As we will see later, one can make good progress on less constrained problems, e.g. related to dissipation, using a slightly extended variational approach (inspired by the point particle example of Section 7). However, we feel that it is useful to first understand the details of the simpler, fully conservative, situation.
 5.
It is important to note the difference between the vorticity formed from the momentum and the corresponding quantity in terms of the velocity. They differ because of the entrainment, and one can show that while the former is conserved along the flow, the latter is not. To avoid confusion we refer to ϖ_{ μν } as the “twist”. This makes some sense because when we use it in Equation (288) we have not yet associated the fourvelocity with the fluid flow.
Notes
Acknowledgments
Several colleagues have helped us develop our understanding of relativistic fluid dynamics. We are particularly indebted to Brandon Carter, David Langlois, Reinhard Prix, and Bernard Schutz.
NA acknowledges support from PPARC via grant no. PPA/G/S/2002/00038 and Senior Research Fellowship no. PP/C505791/1. GLC acknowledges support from NSF grant no. PHY0457072.
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