Go to previous page Go up Go to next page

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 μ U. As mentioned above, the simplest fluid is one for which there is only one four-velocity μ u. Hence, there is a preferred frame defined by uμ, and if we want the observer to sit in this frame we can simply take U μ = uμ. With respect to the fluid there will be no momentum flux, i.e. 𝒫 μ = 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 uμ and the metric gμν. Furthermore, because the spatial stress must also be symmetric, the only possibility is a linear combination of gμν and μ ν u u. Given that μ u 𝒮μν = 0, we find

1- 𝒮μν = 3 𝒮(gμν + uμu ν). (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 = 𝒮∕3 and
T = (ρ + p)u u + pg , (91 ) μν μ ν μν
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 μ ∇μT ν = 0, which follows immediately from the Einstein equations and the fact that μ ∇ μG ν = 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

∂ρ d ρ = ---dn ≡ μ dn, (92 ) ∂n
and we see from the Euler relation (79View Equation) that
μn = p + ρ. (93 )

Let us now get rid of the free index of ∇ μT μν = 0 in two ways: first, by contracting it with uν and second, by projecting it with ⊥ ν ρ (letting Uμ = uμ). Recalling the fact that uμu μ = − 1 we have the identity

ν ν ∇ μ(u u ν) = 0 ⇒ uν∇ μu = 0. (94 )
Contracting with uν and using this identity gives
uμ∇ ρ + (ρ + p)∇ uμ = 0. (95 ) μ μ
The definition of the chemical potential μ and the Euler relation allow us to rewrite this as
μu μ∇ n + μn ∇ uμ ⇒ ∇ n μ = 0, (96 ) μ μ μ
where nμ ≡ nu μ. Projection of the free index using ⊥ ρμ leads to
Dμp = − (ρ + p)aμ, (97 )
where D μ ≡ ⊥ ρ ∇ρ μ is a purely spatial derivative and aμ ≡ uν∇ νuμ 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 μ ∇ μT ν = 0. There is an alternative form that makes the perfect fluid have much in common with vacuum electromagnetism. If we define

μ = μu , (98 ) ν ν
and note that u μduμ = 0, then
dρ = − μ ν dnν. (99 )
The stress-energy-momentum tensor can now be written in the form
T μ = pδμ + n μμ . (100 ) ν ν ν
We have here our first encounter with the fluid element momentum μν that is conjugate to the particle number density current nμ. 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 ∇ Tμ = 0 μ ν using ⊥ μ ν, as before, we find
μ fν + (∇ μn )μ ν = 0, (101 )
where the force density fν is
fν = n μωμν, (102 )
and the vorticity ω μν is defined as
ω ≡ 2∇ μ = ∇ μ − ∇ μ . (103 ) μν [μ ν] μ ν ν μ
Contracting Equation (101View Equation) with nν implies (since ωμν = − ωνμ) that
∇ n μ = 0 (104 ) μ
and as a consequence
f = n μω = 0. (105 ) ν μν
The vorticity two-form ω μν has emerged quite naturally as an essential ingredient of the fluid dynamics [7219Jump To The Next Citation Point1160]. 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 [66Jump To The Next Citation Point]) and is at the heart of turbulence modeling [93].

To demonstrate the role of ωμν as the vorticity, consider a small region of the fluid where the time direction tμ, in local Minkowski coordinates, is adjusted to be the same as that of the fluid four-velocity so that uμ = tμ = (1,0,0, 0). Equation (105View Equation) and the antisymmetry then imply that ω μν can only have purely spatial components. Because the rank of ωμν is two, there are two “nulling” vectors, meaning their contraction with either index of ωμν yields zero (a condition which is true also for vacuum electromagnetism). We have arranged already that tμ be one such vector. By a suitable rotation of the coordinate system the other one can be taken as z μ = (0,0,0,1), thus implying that the only non-zero component of ωμν is ωxy. “MTW” [80Jump To The Next Citation Point] 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 (105View Equation), the vector does not pierce the walls. This is illustrated in Figure 6View Image, 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 6View Image. 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.

View Image

Figure 6: A local, geometrical view of the Euler equation as an integrability condition of the vorticity for a single-constituent perfect fluid.

As we have just seen, the form Equation (105View Equation) 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 kρ. That is, we have

ℒk μρ = 0, −→ (∇ ρμ σ − ∇ σμ ρ)kσ = ∇ ρ(kσμσ). (106 )
Here one may consider two particular situations. If σ k is taken to be the four-velocity, then the scalar σ k μσ represents the “energy per particle”. If instead σ k 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 kσ unspecified, but it is still useful to keep these possibilities in mind. Now contract the equation of motion (105View Equation) with kσ and assume that the conservation law Equation (104View Equation) holds. Then it is easy to show that we have
n ρ∇ ρ (μσkσ) = ∇ ρ(n ρμ σkσ) = 0. (107 )
In other words, we have shown that nρμ kσ σ is a conserved quantity.

Given that we have just inferred the equations of motion from the identity that μ ∇ μT ν = 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 ∇ μT μν = 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.

  Go to previous page Go up Go to next page