Relativistic Fluid Dynamics: Physics for Many Different Scales

There is an extensive literature on special and general relativity, and the spacetime-based view² 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 up-to-date 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 Michaelson-Morley result, the prediction and subsequent discovery of anti-matter, 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 = mc².

In support of general relativity, there are Eötvös-type experiments testing the equivalence of inertial and gravitational mass, detection of gravitational red-shifts of photons, the passing of the solar system tests, confirmation of energy loss via gravitational radiation in the Hulse-Taylor 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!

2.2 Parallel transport and the covariant derivative

In order to have a generally covariant prescription for fluids, i.e. written in terms of spacetime tensors, we must have a notion of derivative that is itself covariant. For example, when acts on a vector V ^ν a rank-two tensor of mixed indices must result:

$${\bar \nabla _\mu}{\bar V^\nu} = {{\partial {x^\sigma}} \over {\partial {{\bar x}^\mu}}}{{\partial {{\bar x}^\nu}} \over {\partial {x^\rho}}}{\nabla _\sigma}{V^\rho}.$$

(13)

The ordinary partial derivative does not work because under a general coordinate transformation

$${{\partial {{\bar V}^\nu}} \over {\partial {{\bar x}^\mu}}} = {{\partial {x^\sigma}} \over {\partial {{\bar x}^\mu}}}{{\partial {{\bar x}^\nu}} \over {\partial {x^\rho}}}{{\partial {V^\rho}} \over {\partial {x^\sigma}}} + {{\partial {x^\sigma}} \over {\partial {{\bar x}^\mu}}}{{{\partial ^2}{{\bar x}^\nu}} \over {\partial {x^\sigma}\partial {x^\rho}}}{V^\rho}.$$

(14)

The second term spoils the general covariance, since it vanishes only for the restricted set of rectilinear transformations

$${\bar x^\mu} = a_\nu ^\mu {x^\nu} + {b^\mu},$$

(15)

where \(a_\nu ^\mu\) and b ^μ are constants.

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 three-dimensional 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.

Consider the classic example of a vector on the surface of a sphere (illustrated in Figure 2). Take that vector and move it along some great circle from the equator to the North pole in such a way as to always keep the vector pointing along the circle. Pick a different great circle, and without allowing the vector to rotate, by forcing it to maintain the same angle with the locally straight portion of the great circle that it happens to be on, move it back to the equator. Finally, move the vector in a similar way along the equator until it gets back to its starting point. The vector’s spatial orientation will be different from its original direction, and the difference is directly related to the particular path that the vector followed.

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On the other hand, we could consider that the sphere is embedded in a three-dimensional Euclidean space, and that the two-dimensional vector on the sphere results from projection of a three-dimensional vector. Move the projection so that its higher-dimensional 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 Levi-Civita [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 force-free, point particle velocity in Euclidean three-dimensional 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 well-versed 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.

We are all well aware that the points on the curve traced out by the particle can be described, in Cartesian coordinates, by three functions x ⁱ (t) where t is the standard Newtonian time. Likewise, we know that the tangent vector at each point of the curve is given by the velocity components v ⁱ (t) = dx ⁱ /dt, and that the force-free condition is equivalent to

$${a^i}(t) = {{{\rm{d}}{v^i}} \over {{\rm{d}}t}} = 0\qquad \Rightarrow \qquad {v^i}(t) = {\rm{const}}.$$

(16)

Hence, the velocity components v ⁱ (0) at the point x ⁱ (0) are equal to those at any other point along the curve, say v ⁱ (T) at x ⁱ (T), and so we could simply take v ⁱ (0) = v ⁱ (T) as the mapping. But as Pauli warns, we only need to reconsider this example using spherical coordinates to see that the velocity components \(\{\dot r,\dot \theta, \dot \phi \}\) must necessarily change as they undergo parallel transport along a straight-line path (assuming the particle does not pass through the origin). The question is what should be used in place of component equality? The answer follows once we find a curvilinear coordinate version of dv ⁱ /dt = 0.

What we need is a new “time” derivative \(\bar D/{\rm{d}}t\), that yields a generally covariant statement

$${{\overline {\rm{D}} {{\bar v}^i}} \over {{\rm{d}}t}} = 0,$$

(17)

where the \({{\bar \upsilon}^i}(t) = {\rm{d}}{{\bar x}^i}/{\rm{d}}t\) are the velocity components in a curvilinear system of coordinates. Consider now a coordinate transformation to the curvilinear coordinate system \({{\bar x}^i}\), the inverse being \({x^i} = {x^i}({{\bar x}^i})\). Given that

$${v^i} = {{\partial {x^i}} \over {\partial {{\bar x}^j}}}{\bar v^j}$$

(18)

we can write

$${{{\rm{d}}{v^i}} \over {{\rm{d}}t}} = \left({{{\partial {x^i}} \over {\partial {{\bar x}^j}}}{{\partial {{\bar v}^j}} \over {\partial {{\bar x}^k}}} + {{{\partial ^2}{x^i}} \over {\partial {{\bar x}^k}\partial {{\bar x}^j}}}{{\bar v}^j}} \right){\bar v^k},$$

(19)

where

$${{{\rm{d}}{{\bar v}^i}} \over {{\rm{d}}t}} = {{\partial {{\bar v}^i}} \over {\partial {{\bar x}^j}}}{\bar v^j}.$$

(20)

Again, we have an “offending” term that vanishes only for rectilinear coordinate transformations. However, we are now in a position to show the importance of this term to a definition of covariant derivative.

First note that the metric for our curvilinear coordinate system is obtainable from

$${\bar g_{ij}} = {{\partial {x^k}} \over {\partial {{\bar x}^i}}}{{\partial {x^l}} \over {\partial {{\bar x}^j}}}{\delta _{kl}},$$

(21)

where

$${\delta _{ij}} = \left\{{\begin{array}{*{20}c} {\rm{1}} & {{\rm{for}}\;i = j,}\\ {\rm{0}} & {{\rm{for}}\;i \neq j.}\\ \end{array}} \right.$$

(22)

Differentiating Equation (21) with respect to \({\bar x}\), and permuting indices, we can show that

$${{{\partial ^2}{x^h}} \over {\partial {{\bar x}^i}\partial {{\bar x}^j}}}{{\partial {x^l}} \over {\partial {{\bar x}^k}}}{\delta _{hl}} = {1 \over 2}({{{\bar g}_{ik,j}} + {{\bar g}_{jk,i}} - {{\bar g}_{ij,k}}}) \equiv {\bar g_{il}}\overline {\left\{{\begin{array}{*{20}c} l \\ {j\;\;k} \\ \end{array}} \right\}} ,$$

(23)

where

$${\bar g_{ij,k}} \equiv {{\partial {{\bar g}_{ij}}} \over {\partial {{\bar x}^k}}}.$$

(24)

Using the inverse transformation of \({{\bar g}_{ij}}\) to δ _ij implied by Equation (21), and the fact that

$${\delta ^i}_j = {{\partial {{\bar x}^k}} \over {\partial {x^j}}}{{\partial {x^i}} \over {\partial {{\bar x}^k}}},$$

(25)

we get

$${{{\partial ^2}{x^i}} \over {\partial {{\bar x}^j}\partial {{\bar x}^k}}} = \overline {{\{_j}{l_{\;k}}\}} {{\partial {x^i}} \over {\partial {{\bar x}^l}}}.$$

(26)

Now we subsitute Equation (26) into Equation (19) and find

$${{{\rm{d}}{v^i}} \over {{\rm{d}}t}} = {{\partial {x^i}} \over {\partial {{\bar x}^j}}}{{\overline {\rm{D}} {{\bar v}^j}} \over {{\rm{d}}t}},$$

(27)

where

$${{\overline {\rm{D}} {{\bar v}^i}} \over {{\rm{d}}t}} = {\bar v^j}\left({{{\partial {{\bar v}^i}} \over {\partial {{\bar x}^j}}} + \overline {{\{_k}{i_j}\}} {{\bar v}^k}} \right).$$

(28)

The operator \(\bar D/{\rm{d}}t\) is easily seen to be covariant with respect to general transformations of curvilinear coordinates.

We now identify our generally covariant derivative (dropping the overline) as

$${\nabla _j}{v^i} = {{\partial {v^i}} \over {\partial {x^j}}} + {\{_k}{i_j}\} {v^k} \equiv {v^i}_{;j}.$$

(29)

Similarly, the covariant derivative of a covector is

$${\nabla _j}{v_i} = {{\partial {v_i}} \over {\partial {x^j}}} - {\{_i}{k_j}\} {v_k} \equiv {v_{i;j}}.$$

(30)

One extends the covariant derivative to higher rank tensors by adding to the partial derivative each term that results by acting linearly on each index with \(\left\{{\begin{array}{*{20}c} {\;i} \\ {k\;\;j} \\ \end{array}} \right\}\) using the two rules given above.

By relying on our understanding of the force-free point particle, we have built a notion of parallel transport that is consistent with our intuition based on equality of components in Cartesian coordinates. We can now expand this intuition and see how the vector components in a curvilinear coordinate system must change under an infinitesimal, parallel displacement from x ⁱ (t) to x ⁱ (t+δt). Setting Equation (28) to zero, and noting that v ⁱ δt = δx ⁱ , implies

$$\delta {v^i} \equiv {{\partial {v^i}} \over {\partial {x^j}}}\delta {x^j} = - {\{_k}{i_j}\} {v^k}\delta {x^j}.$$

(31)

In general relativity we assume that under an infinitesimal parallel transport from a spacetime point x ^μ (λ) on a given curve to a nearby point x ^μ (λ + δλ) on the same curve, the components of a vector will change in an analogous way, namely

$$\delta V_\parallel ^\mu \equiv {{\partial {V^\mu}} \over {\partial {x^\nu}}}\delta {x^\nu} = - \Gamma _{\rho \nu}^\mu {V^\rho}\delta {x^\nu},$$

(32)

where

$$\delta {x^\mu} \equiv {{{\rm{d}}{x^\mu}} \over {{\rm{d}}\lambda}}\delta \lambda .$$

(33)

Weyl [118] refers to the symbol \(\Gamma _{\rho \nu}^\mu\) as the “components of the affine relationship”, but we will use the modern terminology and call it the connection. In the language of Weyl and Pauli, this is the mapping that we were looking for.

For Euclidean space, we can verify that the metric satisfies

$${\nabla _i}{g_{jk}} = 0$$

(34)

for a general, curvilinear coordinate system. The metric is thus said to be “compatible” with the covariant derivative. We impose metric compatibility in general relativity. This results in the so-called Christoffel symbol for the connection, defined as

$$\Gamma _{\mu \nu}^\rho = {1 \over 2}{g^{\rho \sigma}}({{g_{\mu \sigma ,\nu}} + {g_{\nu \sigma ,\mu}} - {g_{\mu \nu ,\sigma}}}){.}$$

(35)

The rules for the covariant derivative of a contravariant vector and a covector are the same as in Equations (29) and (30), except that all indices are for full spacetime.

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, non-relativistic 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 Chandrasekhar-Friedman-Schutz 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 coordinate-based discussion of Schouten [99], as it may be more readily understood by those not well-versed 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 axis-orientations 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 “push-forward” the vector, and then subject it to a “pull-back”.

In the active, push-forward sense we imagine that there are two spacetime points connected by a smooth curve x ^μ (δ). Let the first point be at λ = 0, and the second, nearby point at λ = ϵ, i.e. xμ(ϵ); that is,

$$x_\epsilon ^\mu \equiv {x^\mu}(\epsilon) \approx x_0^\mu + \epsilon \,{\xi ^\mu},$$

(36)

where \(x_0^\mu = {x^\mu}(0)\) and

$${\xi ^\mu} = {\left. {{{{\rm{d}}{x^\mu}} \over {{\rm{d}}\lambda}}} \right\vert _{\lambda = 0}}$$

(37)

is the tangent to the curve at λ = 0. In the passive, pull-back sense we imagine that the coordinate system itself is changed to \({{\bar x}^\mu} = {{\bar x}^\mu}({x^\nu})\), but in the very special form

$${\bar x^\mu} = {x^\mu} - \epsilon {\xi ^\mu}.$$

(38)

It is in this last step that the Lie derivative differs from the covariant derivative. In fact, if we insert Equation (36) into Equation (38) we find the result \(\bar x_\epsilon ^\mu = x_0^\mu\). This is called “Lie-dragging” of the coordinate frame, meaning that the coordinates at λ = 0 are carried along so that at λ = ϵ, and in the new coordinate system, the coordinate labels take the same numerical values.

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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 high-test Zebco parallel transport fishing pole, the Lie derivative with the Shimano, Lie-dragging ultra-light. Let us now see how Lie-dragging reels in vectors.

For some given vector field that takes values V ^μ (λ), say, along the curve, we write

$$V_0^\mu = {V^\mu}(0)$$

(39)

for the value of V ^μ at λ = 0 and

$$V_\epsilon ^\mu = {V^\mu}(\epsilon)$$

(40)

for the value at λ = ϵ. Because the two points \(x_0^\mu\) and \(x_\epsilon ^\mu\) are infinitesimally close (ϵ ≪ 1) and we can thus write

$$V_\epsilon ^\mu \approx V_0^\mu + \epsilon \,{\xi ^\nu}{\left. {{{\partial {V^\mu}} \over {\partial {x^\nu}}}} \right\vert _{\lambda = 0}}$$

(41)

for the value of V ^μ at the nearby point and in the same coordinate system. However, in the new coordinate system at the nearby point, we find

$$\bar V_\epsilon ^\mu = {\left. {\left({{{\partial {{\bar x}^\mu}} \over {\partial {x^\nu}}}{V^\nu}} \right)} \right\vert _{\lambda = \epsilon}} \approx V_\epsilon ^\mu - \epsilon \,V_0^\nu {\left. {{{\partial {\xi ^\mu}} \over {\partial {x^\nu}}}} \right\vert _{\lambda = 0}}.$$

(42)

The Lie derivative now is defined to be

$$\begin{array}{*{20}c} {{{\mathcal L}_\xi}{V^\mu} = \underset{\epsilon \rightarrow 0}{\lim} {{\bar V_\epsilon ^\mu - {V^\mu}} \over \epsilon}\quad \quad \quad \quad \;\;\;}\\ {= {\xi ^\nu}{{\partial {V^\mu}} \over {\partial {x^\nu}}} - {V^\nu}{{\partial {\xi ^\mu}} \over {\partial {x^\nu}}}}\;\;\\ {= {\xi ^\nu}{\nabla _\nu}{V^\mu} - {V^\nu}{\nabla _\nu}{\xi ^\mu},}\\ \end{array}$$

(43)

where we have dropped the “0” subscript and the last equality follows easily by noting \(\Gamma _{\mu \nu}^\rho = \Gamma _{\nu \mu}^{^\rho}\).

The Lie derivative of a covector A _μ is easily obtained by acting on the scalar A _μ V ^μ for an arbitrary vector V ^μ :

$$\begin{array}{*{20}c} {{{\mathcal L}_\xi}{A_\mu}{V^\mu} = {V^\mu}{{\mathcal L}_\xi}{A_\mu} + {A_\mu}{{\mathcal L}_\xi}{V^\mu}\quad \quad \quad \quad \quad \quad \quad \quad \quad\;}\\ {= {V^\mu}{{\mathcal L}_\xi}{A_\mu} + {A_\mu}({{\xi ^\nu}{\nabla _\nu}{V^\mu} - {V^\nu}{\nabla _\nu}{\xi ^\mu}}){.}}\\ \end{array}$$

(44)

But, because A _μ V ^μ is a scalar,

$$\begin{array}{*{20}c} {{{\mathcal L}_\xi}{A_\mu}{V^\mu} = {\xi ^\nu}{\nabla _\nu}{A_\mu}{V^\mu}\quad \quad \quad \quad \quad \quad \quad \quad \quad}\\ {= {\xi ^\nu}({{V^\mu}{\nabla _\nu}{A_\mu} + {A_\mu}{\nabla _\nu}{V^\mu}}),}\\ \end{array}$$

(45)

and thus

$${V^\mu}({{{\mathcal L}_\xi}{A_\mu} - {\xi ^\nu}{\nabla _\nu}{A_\mu} - {A_\nu}{\nabla _\mu}{\xi ^\nu}}) = 0{.}$$

(46)

Since V ^μ is arbitrary,

$${{\mathcal L}_\xi}{A_\mu} = {\xi ^\nu}{\nabla _\nu}{A_\mu} + {A_\nu}{\nabla _\mu}{\xi ^\nu}.$$

(47)

We introduced in Equation (32) the effect of parallel transport on vector components. By contrast, the Lie-dragging of a vector causes its components to change as

$$\delta V_{\mathcal L}^\mu = {{\mathcal L}_\xi}{V^\mu}\,\epsilon .$$

(48)

We see that if ℒ _ξ V ^μ = 0, then the components of the vector do not change as the vector is Liedragged. Suppose now that V ^μ represents a vector field and that there exists a corresponding congruence of curves with tangent given by ξ ^μ . If the components of the vector field do not change under Lie-dragging we can show that this implies a symmetry, meaning that a coordinate system can be found such that the vector components will not depend on one of the coordinates. This is a potentially very powerful statement.

Let ξ ^μ represent the tangent to the curves drawn out by, say, the μ = a coordinate. Then we can write x ^a (δ) = δ which means

$${\xi ^\mu} = {\delta ^\mu}_a.$$

(49)

if the Lie derivative of V ^μ with respect to ξ ^ν vanishes we find

$${\xi ^\nu}{{\partial {V^\mu}} \over {\partial {x^\nu}}} = {V^\nu}{{\partial {\xi ^\mu}} \over {\partial {x^\nu}}} = 0.$$

(50)

Using this in Equation (41) implies \(V_\epsilon ^\mu = V_0^\mu\), that is to say, the vector field V ^μ (x ^ν ) does not depend on the x ^a coordinate. Generally speaking, every ξ ^μ that exists that causes the Lie derivative of a vector (or higher rank tensors) to vanish represents a symmetry.

Let us take the spacetime metric g _μν as an example. A spacetime symmetry can be represented by a generating vector field ξ ^μ such that

$${{\mathcal L}_\xi}{g_{\mu \nu}} = {\nabla _\mu}{\xi _\nu} + {\nabla _\nu}{\xi _\mu} = 0.$$

(51)

This is known as Killing’s equation, and solutions to this equation are naturally referred to as Killing vectors. As a particular case, let us consider the class of stationary, axisymmetric, and asymptotically flat spacetimes. These are highly relevant in the present context since they capture the physics of rotating, equilibrium configurations. In other words, the geometries that result are among the most fundamental, and useful, for the relativistic astrophysics of spinning black holes and neutron stars. Stationary, axisymmetric, and asymptotically flat spacetimes are such that [14]

1. 1.

   there exists a Killing vector t ^μ that is timelike at spatial infinity;

    
2. 2.

   there exists a Killing vector ϕ ^μ that vanishes on a timelike 2-surface (called the axis of symmetry), is spacelike everywhere else, and whose orbits are closed curves; and

    
3. 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.

    

These conditions imply [16] that a coordinate system exists where

$${t^\mu} = (1,0,0,0),\qquad {\phi ^\mu} = (0,0,0,1){.}$$

(52)

So, for instance, the two Lie derivatives of the metric g _μν , say, are such that

$$\begin{array}{*{20}c} {{{\mathcal L}_t}{g_{\mu \nu}} = {t^\sigma}{\nabla _\sigma}{g_{\mu \nu}} + {g_{\mu \sigma}}{\nabla _\nu}{t^\sigma} + {g_{\sigma \nu}}{\nabla _\mu}{t^\sigma}}\\ {= {g_{\mu \sigma}}{\partial _\nu}{t^\sigma} + {g_{\sigma \nu}}{\partial _\mu}{t^\sigma}\quad \quad}\\ {= 0,\quad \quad \quad \quad \quad \quad \quad \quad\;\;}\\ \end{array}$$

(53)

and

$$\begin{array}{*{20}c} {{{\mathcal L}_\phi}{g_{\mu \nu}} = {\phi ^\sigma}{\nabla _\sigma}{g_{\mu \nu}} + {g_{\mu \sigma}}{\nabla _\nu}{\phi ^\sigma} + {g_{\sigma \nu}}{\nabla _\mu}{\phi ^\sigma}}\\ {= {g_{\mu \sigma}}{\partial _\nu}{\phi ^\sigma} + {g_{\sigma \nu}}{\partial _\mu}{\phi ^\sigma}\quad \quad}\\ {= 0.\quad \quad \quad \quad \quad \quad \quad \quad \;\;\;}\\ \end{array}$$

(54)

Without an a priori, physically-based 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 G₀₀ and the pressure to be G₁₁, 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 non-vacuum spacetime. As such, the strategy is antithetical to the raison d’être of gravitational-wave astrophysics, which is to use gravitational-wave 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 gravitational-wave emission from realistic neutron stars. To achieve this aim, we need an appreciation of the stress-energy 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. well-behaved 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).

To understand what the various components of T _μν mean physically we will write them in terms of projections into the timelike and spacelike directions associated with a given observer. In order to project a tensor index along the observer’s timelike direction we contract that index with the observer’s unit four-velocity U ^μ . A projection of an index into spacelike directions perpendicular to the timelike direction defined by U ^μ (see [105] for the idea from a “3 + 1” point of view, or [21] from the “brane” point of view) is realized via the operator defined as

$$\bot _\nu ^\mu = {\delta ^\mu}_\nu + {U^\mu}{U_\nu},\qquad {U^\mu}{U_\mu} = - 1.$$

(66)

Any tensor index that has been “hit” with the projection operator will be perpendicular to the timelike direction associated with U ^μ in the sense that \(\bot _\nu ^\mu {U^\nu} = 0\). Figure 4 is a local view of both projections of a vector V ^μ for an observer with unit four-velocity U ^μ . More general tensors are projected by acting with U ^μ or \(\bot _\nu ^\mu\) on each index separately (i.e. multi-linearly).

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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 D that 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. Strictly speaking, our model has L infinitesimal, M → ∞, 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.

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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 He⁴ [94] at non-zero temperature and a mixture of superfluid He⁴ and He³ [8].

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 so-called 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.

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 “off-the-shelve” 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 “off-the-shelve” approach). One’s footing is then always made sure by well-grounded, action-based 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 well-posed action is, of course, perfect for systematically constructing the momenta.

6.2 The single, perfect fluid problem: “Off-the-shelf” consistency analysis

We earlier took the components of a general stress-energy-momentum tensor and projected them onto the axes of a coordinate system carried by an observer moving with four-velocity U ^μ . As mentioned above, the simplest fluid is one for which there is only one four-velocity 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 stress-energy-momentum tensor will transform properly under general coordinate transformations, and hence can be used for any observer.

The spatial stress is a two-index, symmetric tensor, and the only objects that can be used to carry the indices are the four-velocity u ^μ and the metric g _μν . Furthermore, because the spatial stress must also be symmetric, the only possibility is a linear combination of g _μν and u ^μ u ^ν . Given that \({u^\mu}{{\mathcal S}_{\mu \nu}} = 0\), we find

$${{\mathcal S}_{\mu \nu}} = {1 \over 3}{\mathcal S}({g_{\mu \nu}} + {u_\mu}{u_\nu}){.}$$

(90)

As the system is assumed to be locally isotropic, it is possible to diagonalize the spatial-stress tensor. This also implies that its three independent diagonal elements should actually be equal to the same quantity, which is the local pressure. Hence we have \(p = {\mathcal S}/3\) and

$${T_{\mu \nu}} =({\rho + p}){u_\mu}{u_\nu} + p{g_{\mu \nu}},$$

(91)

which is the well-established result for a perfect fluid.

Given a relation p = p(ρ), there are four independent fluid variables. Because of this the equations of motion are often understood to be \({\nabla _\mu}{T^\mu}_\nu = 0\), which follows immediately from the Einstein equations and the fact that \({\nabla _\mu}{G^\mu}_\nu = 0\). To simplify matters, we take as equation of state a relation of the form ρ = ρ(n) where n is the particle number density. The chemical potential μ is then given by

$${\rm{d}}\rho = {{\partial \rho} \over {\partial n}}{\rm{d}}n \equiv \mu \,{\rm{d}}n,$$

(92)

and we see from the Euler relation (79) that

$$\mu n = p + \rho .$$

(93)

Let us now get rid of the free index of \({\nabla _\mu}{T^\mu}_\nu = 0\) in two ways: first, by contracting it with u ^ν and second, by projecting it with \(\bot _\rho ^\nu\) (letting U ^μ = u ^μ ). Recalling the fact that U ^μ u _μ = − 1 we have the identity

$${\nabla _\mu}({{u^\nu}{u_\nu}}) = 0\qquad \Rightarrow \qquad {u_\nu}{\nabla _\mu}{u^\nu} = 0.$$

(94)

Contracting with u ^ν and using this identity gives

$${u^\mu}{\nabla _\mu}\rho + (\rho + p){\nabla _\mu}{u^\mu} = 0.$$

(95)

The definition of the chemical potential μ and the Euler relation allow us to rewrite this as

$$\mu {u^\mu}{\nabla _\mu}n + \mu n{\nabla _\mu}{u^\mu}\qquad \Rightarrow \qquad {\nabla _\mu}{n^\mu} = 0,$$

(96)

where n ^μ = nu ^μ . Projection of the free index using \(\bot _\mu ^\rho\) leads to

$${D_\mu}p = - (\rho + p){a_\mu},$$

(97)

where \({D_\mu} \equiv \bot _\mu ^\rho {\nabla _\rho}\) is a purely spatial derivative and a _μ ≡ u ^ν ∇ _ν u _μ is the acceleration. This is reminiscent of the familiar Euler equation for Newtonian fluids.

However, we should not be too quick to think that this is the only way to understand \({\nabla _\mu}{T^\mu}_\nu = 0\). There is an alternative form that makes the perfect fluid have much in common with vacuum electromagnetism. If we define

$${\mu _\nu} = \mu {u_\nu},$$

(98)

and note that u _μ d_u ^μ = 0, then

$${\rm{d}}\rho = - {\mu _\nu}\,{\rm{d}}{n^\nu}.$$

(99)

The stress-energy-momentum tensor can now be written in the form

$${T^\mu}_\nu = p{\delta ^\mu}_\nu + {n^\mu}{\mu _\nu}.$$

(100)

We have here our first encounter with the fluid element momentum μ _ν that is conjugate to the particle number density current n ^μ . Its importance will become clearer as this review develops, particularly when we discuss the two-fluid case. If we now project onto the free index of \({\nabla _\mu}{T^\mu}_\nu = 0\) using \(\bot _\nu ^\mu\), as before, we find

$${f_\nu} + \left({{\nabla _\mu}{n^\mu}} \right){\mu _\nu} = 0,$$

(101)

where the force density f _ν is

$${f_\nu} = {n^\mu}{\omega _{\mu \nu}},$$

(102)

and the vorticity ω _μν is defined as

$${\omega _{\mu \nu}} \equiv 2\nabla {}_{\left[ \mu \right.}{\mu _{\left. \nu \right]}} = {\nabla _\mu}{\mu _\nu} - {\nabla _\nu}{\mu _\mu}.$$

(103)

Contracting Equation (101) with n ^ν implies (since ω _μν = −ω _νμ ) that

$${\nabla _\mu}{n^\mu} = 0$$

(104)

and as a consequence

$${f_\nu} = {n^\mu}{\omega _{\mu \nu}} = 0.$$

(105)

The vorticity two-form ω _μν has emerged quite naturally as an essential ingredient of the fluid dynamics [72, 19, 11, 60]. Those who are familiar with Newtonian fluids should be inspired by this, as the vorticity is often used to establish theorems on fluid behavior (for instance the Kelvin-Helmholtz theorem [66]) and is at the heart of turbulence modeling [93].

To demonstrate the role of as the vorticity, consider a small region of the fluid where the time direction t ^μ , in local Minkowski coordinates, is adjusted to be the same as that of the fluid four-velocity so that u ^μ = t ^μ = (1, 0, 0, 0). Equation (105) and the antisymmetry then imply that ω _μν can only have purely spatial components. Because the rank of ω _μν is two, there are two “nulling” vectors, meaning their contraction with either index of ω _μν yields zero (a condition which is true also for vacuum electromagnetism). We have arranged already that t ^μ be one such vector. By a suitable rotation of the coordinate system the other one can be taken as z ^u = (0, 0, 0, 1), thus implying that the only non-zero component of ω _μν is x _μν . “MTW” [80] points out that such a two-form can be pictured geometrically as a collection of oriented worldtubes, whose walls lie in the x = const and y = const planes. Any contraction of a vector with a two-form that does not yield zero implies that the vector pierces the walls of the worldtubes. But when the contraction is zero, as in Equation (105), the vector does not pierce the walls. This is illustrated in Figure 6, where the red circles indicate the orientation of each world-tube. The individual fluid element four-velocities lie in the centers of the world-tubes. Finally, consider the closed contour in Figure 6. If that contour is attached to fluid-element worldlines, then the number of worldtubes contained within the contour will not change because the worldlines cannot pierce the walls of the worldtubes. This is essentially the Kelvin-Helmholtz theorem on conservation of vorticity. From this we learn that the Euler equation is an integrability condition which ensures that the vorticity two-surfaces mesh together to fill out spacetime.

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As we have just seen, the form Equation (105) of the equations of motion can be used to discuss the conservation of vorticity in an elegant way. It can also be used as the basis for a derivation of other known theorems in fluid mechanics. To illustrate this, let us derive a generalized form of Bernoulli’s theorem. Let us assume that the flow is invariant with respect to transport by some vector field k ^ρ . That is, we have

$${{\mathcal L}_k}{\mu _\rho} = 0,\quad \quad \rightarrow \quad \quad ({\nabla _\rho}{\mu _\sigma} - {\nabla _\sigma}{\mu _\rho}){k^\sigma} = {\nabla _\rho}({k^\sigma}{\mu _\sigma}).$$

(106)

Here one may consider two particular situations. If k ^σ is taken to be the four-velocity, then the scalar k ^σ μ _σ represents the “energy per particle”. If instead k ^σ represents an axial generator of rotation, then the scalar will correspond to an angular momentum. For the purposes of the present discussion we can leave k ^σ unspecified, but it is still useful to keep these possibilities in mind. Now contract the equation of motion (105) with k ^σ and assume that the conservation law Equation (104) holds. Then it is easy to show that we have

$${n^\rho}{\nabla _\rho}\left({{\mu _\sigma}{k^\sigma}} \right) = {\nabla _\rho}\left({{n^\rho}{\mu _\sigma}{k^\sigma}} \right) = 0.$$

(107)

In other words, we have shown that n ^ρ μ _σ k ^σ is a conserved quantity.

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 multi-fluid case: Then the vanishing of the covariant divergence represents only four equations, whereas the multi-fluid 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 “off-the-shelf” strategy is concerned, and now move on to an action-based derivation of the fluid equations of motion.

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 action-based 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.

In this section the equations of motion and the stress-energy-momentum tensor for a one-component, general relativistic fluid are obtained from an action principle. Specifically a so-called “pull-back” approach (see, for instance, [36, 37, 34]) is used to construct a Lagrangian displacement of the number density four-current 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.

As there is only one species of particle considered here, n ^μ is conserved, meaning that once a number of particles N is assigned to a particular fluid element, then that number is the same at each point of the fluid element’s worldline. This would correspond to attaching a given number of particles (i.e. N₁, N₂, etc.) to each of the worldlines in Figure 7. Mathematically, one can write this as a standard particle-flux conservation equation⁴:

$${\nabla _\mu}{n^\mu} = 0.$$

(121)

for reasons that will become clear shortly, it is useful to rewrite this conservation law in what may (if taken out of context) seem a rather obscure way. We introduce the dual n _{ν λ τ} to n ^μ , i.e.

$${n_{\nu \lambda \tau}} = {\epsilon _{\nu \lambda \tau \mu}}{n^\mu},\quad {n^\mu} = {1 \over {3!}}{\epsilon ^{\mu \nu \lambda \tau}}{n_{\nu \lambda \tau}},$$

(122)

such that

$${n^2} = {1 \over {3!}}{n_{\nu \lambda \tau}}{n^{\nu \lambda \tau}}.$$

(123)

This shows that n _{ν λ τ} acts as a volume measure which, for example, allows us to “count” the number of fluid elements. In Figure 6 we have seen that a two-form gives worldtubes. A three-form is the next higher-ranked object and it can be thought of, in an analogous way, as leading to boxes [80]. The key step here is to realize that the conservation rule is equivalent to having the three-form n _{ν λ τ} be closed. In differential geometry this means that

$$\nabla {}_{\left[ \mu \right.}{n_\nu}{\lambda _{\left. \tau \right]}} = 0.$$

(124)

This can be shown to be equivalent to Equation (121) by contracting with ϵ _{μν λ τ} .

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The main reason for introducing the dual is that it is straightforward to construct a particle number density three-form 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(n²) 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 well-established 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 left-hand-side 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 right-hand-side of Equation (152).

Given that we only have one four-velocity, 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.

Having discussed the single fluid model, and how one accounts for stratification, it is time to move on to the problem of modeling multi-fluid 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 rest-frame. 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 heat-conducting 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 multi-fluid 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.

We make this happen by introducing the three-dimensional matter space, the difference being that we now need two such spaces. These will be labelled by coordinates and we recall that A, B, C, etc. = 1, 2, 3. This is depicted in Figure 8, which indicates the important facts that (i) a given spatial point can be intersected by each fluid’s worldline and (ii) the individual worldlines are not necessarily parallel at the intersection, i.e. the independent fluids are interpenetrating and can exhibit a relative flow with respect to each other. Although we have not indicated this in Figure 8 (in order to keep the figure as uncluttered as possible) attached to each worldline of a given constituent will be a fixed number of particles \(N_1^{\rm{x}}\), \(N_2^{\rm{x}}\), etc. (cf. Figure 7). For the same reason, we have also not labelled (as in Figure 7) the “push-forwards” (represented by the arrows) from the matter spaces to spacetime.

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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 ⁴He and mixtures of superfluid ³He and ⁴He). In the case of neutron stars, entrainment is an essential ingredient of the current best explanation for the so-called glitches [95, 97]. Carter [19] has also argued that these “anomalous” terms are necessary for causally well-behaved heat conduction in relativistic fluids, and by extension necessary for building well-behaved relativistic equations that incorporate dissipation.

It must be noted that Equation (183) is significantly different from the multi-constituent version of Equation (166). This is true even if one is solving for a static and spherically symmetric configuration, where the fluid four-velocities 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 double-constituent star with only one four-velocity.

Crucial to the understanding of black holes is the effect of spacetime curvature on the light-cone structures, that is, the totality of null vectors that emanate from each spacetime point. Crucial to the propagation of massless fields is the light-cone 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 plane-wave propagation to derive the speed(s) of sound in the single-fluid, two-constituent single-fluid, and two-fluid 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 multi-fluids impact the local dynamics.

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 action-based 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 non-relativistic 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 two-fluid 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 He⁴ [94, 110], where the entropy can flow independently of the superfluid Helium atoms. Superfluid He³ can also be included in the mixture, in which case there will be a relative flow of the He³ isotope with respect to He⁴, and relative flows of each with respect to the entropy [113].

Investigations into the stability properties of rotating self-gravitating 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 gravitational-wave signals. In this section we will review the Lagrangian perturbation framework developed by Friedman and Schutz [44, 45] for rotating non-relativistic 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 so-called r-modes at all rotation rates in a perfect fluid star.

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 gravitational-wave driven instability of the fundamental f-mode or the inertial r-mode 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 r-mode 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 non-causality is built into the description is unattractive already in the context of the Navier-Stokes equations. After all, one would expect heat to propagate at roughly the mean molecular speed in the system. For a relativistic description non-causal 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.

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 high-energy 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 high-energy nuclear collision is viewed in the following way: In the center-of-mass 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 nucleon-nucleon 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 “freeze-out” conditions. The problem is complicated by the fact that the “freeze-out” typically occurs at a different time for each fluid cell.

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 freeze-out, 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 heavy-ion collisions are very similar to those faced in studies of relativistic dissipation theory and multi-fluid 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 heavy-ion collisions it is common to choose u ^μ as the velocity of either energy transport (the Landau-Lifshitz 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 Landau-Lifshitz 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 particle-antiparticle pairs. Since the net baryon number in this region vanishes, the Eckart definition of the four-velocity 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 frame-independent description will need to include several distinct fluid components.

Multi-fluid models have, in fact, often been considered for heavy-ion collisions. One can, for example, treat the target and projectile nuclei as separate fluids to admit interpenetration, thus arriving at a two-fluid model. One could also use a relativistic multi-fluid 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 space-time 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 baryon-rich 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 center-of-baryon number (analogous to the center-of-mass) vanishes.

The main problem with the one-fluid 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 mean-free 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 mean-free 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 heavy-ion collisions no calculations have yet been performed using a fully three-dimensional, 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].

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 He⁴ [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 heavy-ion collisions with broken symmetries.

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 multi-fluid 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, gamma-ray 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!