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8 The “Pull-back” Formalism for a Single Fluid

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, [36Jump To The Next Citation Point3734]) is used to construct a Lagrangian displacement of the number density four-current nμ, 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, μ n 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 7View Image. Mathematically, one can write this as a standard particle-flux conservation equation4:

μ ∇ μn = 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 νλτ to n μ, i.e. 
μ μ 1- μνλτ n νλτ = ενλτμn , n = 3!ε n νλτ, (122 )
such that
2 1 νλτ n = --n νλτn . (123 ) 3!
This shows that nνλτ acts as a volume measure which, for example, allows us to “count” the number of fluid elements. In Figure 6View Image 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 [80Jump To The Next Citation Point]. The key step here is to realize that the conservation rule is equivalent to having the three-form n νλτ be closed. In differential geometry this means that
∇ [μnνλτ] = 0. (124 )
This can be shown to be equivalent to Equation (121View Equation) by contracting with εμνλτ.

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.

View Image

Figure 7: The push-forward from “fluid-particle” points in the three-dimensional matter space labelled by the coordinates {X1, X2, X3 } to fluid-element worldlines in spacetime. Here, the push-forward of the “Ith” (I = 1,2,...,n) fluid-particle to, say, an initial point on a worldline in spacetime can be taken as A A i X I = X (0, xI) where i xI is the spatial position of the intersection of the worldline with the t = 0 time slice.

This can be made to happen by introducing a three-dimensional “matter” space – the left-hand part of Figure 7View Image – which can be labelled by coordinates XA, where A, B,C, etc.= 1,2,3. For each time slice in spacetime, there will be the same configuration in the matter space; that is, as time goes forward, the fluid particle positions in the matter space remain fixed even as the worldlines form their weavings in spacetime. In this sense we are “pushing forward” from the matter space to spacetime (cf. the previous discussion of the Lie derivative). The three-form can be pulled back to its three-dimensional matter space by using the mappings XA. This construction then guarantees a three-form that is automatically closed on spacetime, namely

( A )( B ) C nνλτ = NABC ∇ νX ∇ λX (∇ τX ), (125 )
where NABC is completely antisymmetric in its indices and is a function only of the A X. As implied by Figure 7View Image the XA are also comoving on their respective worldlines, meaning that they are independent of the proper time τ, say, that parameterizes each curve:
dXA 1 ( )( )( ) ( ) n -----≡ nμ∇ μXA = ℒnXA = --εμνλτNBCD ∇ νXB ∇λXC ∇τXD ∇ μXA = 0. (126 ) dτ 3!
The time part of the spacetime dependence of the XA is thus somewhat ad hoc; that is, if we take the flow of time tμ to be the proper time of the worldlines (tμ is parallel to nμ), the XA do not change. An apparent time dependence in spacetime means that tμ is such as to cut across fluid worldlines (μ t is not parallel to μ n), which of course have different values for the XA.

Because the matter space indices are three-dimensional and the closure condition involves four spacetime indices, and also the A X are scalars on spacetime (and thus two covariant differentiations commute), the construction does indeed produce a closed three-form:

( ) ∇ n = ∇ NABC ∇ νXA ∇ λXB ∇ XC ≡ 0. (127 ) [μ νλτ] [μ τ]
In terms of the scalar fields XA, we now have particle number density currents that are automatically conserved. Thus, another way of viewing the construction is that the fundamental fluid field variables are the A X. In fact, the variations of n νλτ can now be taken with respect to variations of the A X, namely
( ) δn = ∂NABC--∇ XA ∇ XB ∇ XC δXD νλτ ∂XD ν λ τ ( A B C A B C A B C ) +NABC ∇ νδX ∇λX ∇ τX + ∇ νX ∇ λδX ∇ τX + ∇ νX ∇ λX ∇ τδX . (128 )
As the A X are scalars, the covariant derivatives can be replaced with ordinary partial derivatives, meaning that there is no metric variation to consider.

Let us introduce the Lagrangian displacement on spacetime for the particles, to be denoted ξμ. This is related to the variation δXA via another push-forward from the matter space into spacetime,

( ) δXA = − ∇ μXA ξ μ = − ℒ ξXA, (129 )
where ℒξ is the Lie derivative along μ ξ (cf. Equation (43View Equation)). Using the fact that
A ([ A ] μ) ∇ νδX = − ∇( ν ∇ μX) ξ ( ) = − ∇ XA ∇ ξμ − ∇ ∇ XA ξμ (130 ) μ ν μ ν
means [67]
δn νλτ = − (ξ σ∇ σnνλτ + nσλτ∇ νξσ + nνστ∇ λξσ + nνλσ∇ τξσ) = − ℒξnνλτ. (131 )
Using Equation (122View Equation) above and Equation (367View Equation) of Appendix A, we can thus infer that
( 1 ) δnμ = n σ∇ σξμ − ξσ∇ σnμ − nμ ∇ σξσ + --gσρδgσρ ( ) 2 = − ℒ ξn μ − nμ ∇ σξσ + 1g σρδg σρ . (132 ) 2
This constraint guarantees that nμ is conserved. By introducing the decomposition
μ μ μ n = nu , uμu = − 1, (133 )
we can furthermore show that
( ) σ σ ν 1- σρ σ ρ δn = − ∇ σ (nξ ) − n u νu ∇ σξ + 2 [g + u u ]δgσρ , (134 )
δuμ = (δμρ + uμu ρ)(uσ∇ σξρ − ξσ∇ σuρ) + 1u μuσuρδg σρ. (135 ) 2

The Lagrangian variation resulting from the Lagrangian displacement is given by

Δ ≡ δ + ℒξ, (136 )
from which it follows that
Δn μλτ = 0, (137 )
which is entirely consistent with the pull-back construction. We also find that
μ 1 μ σ ρ Δu = -u u u Δg σρ, (138 ) 2 Δ ενλτσ = 1ενλτσgμρΔg μρ, (139 ) 2 Δn = − n-(gσρ + uσu ρ)Δg . (140 ) 2 σρ
These formulae and their Newtonian analogues have been adroitly used by Friedman and Schutz in their establishment of the so-called Chandrasekhar–Friedman–Schutz (CFS) instability [3044Jump To The Next Citation Point45Jump To The Next Citation Point] (see Section 13). Later, in Section 12, we take the limit that the speed of light becomes large so as to construct non-relativistic fluid equations. The same procedure can be used to obtain the Newtonian analogues of the formulae constructed here.

At first glance, there appears to be a glaring inconsistency between the pull-back construction and the Lagrangian variation, since the latter seems to have four independent components, but the former clearly has three. In fact, there is a gauge freedom in the Lagrangian variation that can be used to reduce the number of independent components. Take Equation (132View Equation) and substitute

ξμ = ξμ + 𝒢 μ. (141 )
Using the fact that ∇ μnμ = 0, it can be shown that
μ --μ 1 μστλ ρ ν δn = δn − -ε ∇ σ (n ετλρνu 𝒢 ), (142 ) 2
where -- δnμ is as in Equation (132View Equation) except ξμ is replaced with -μ ξ. If we set
𝒢μ = 𝒢u μ, (143 )
then the last term vanishes and μ --μ δn = δn. Thus, we can use the arbitrary function 𝒢 to reduce the number of independent ξμ to three.

With a general variation of the conserved four-current in hand, we can now use an action principle to derive the equations of motion and the stress-energy-momentum tensor. The central quantity in the analysis is the so-called “master” function Λ, which is a function of the scalar 2 μ n = − nμn. For this single fluid system, it is such that − Λ corresponds to the local thermodynamic energy density. In the action principle, the master function is the Lagrangian density for the fluid, i.e. for a spacetime region ℳ the fluid action is

∫ √ --- 4 Ifluid = ℳ − g Λ d x. (144 )
To obtain the Einstein equation (65View Equation) we must add the Einstein–Hilbert action:
∫ ∫ -1-- √ --- 4 -1- √-- 3 IEH = 16π ℳ − g R d x + 8π ∂ℳ h K d x, (145 )
where R is the Ricci scalar, h is the determinant of the metric of the boundary ∂ℳ (if present) of ℳ, and K is the trace of the extrinsic curvature of the boundary. This last term is added to ensure that fixing the metric on the boundary leads to a well-posed action principle [124].

We should point out that our consideration of a master function of the form 2 Λ = Λ (n ) 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 Aμ to form the additional scalar n μAμ (as would be the case with coupling to electromagnetism, say). A term that could be added is of the form n μ∇ φ μ for an arbitrary scalar field φ. Unlike the previous two possible additional terms, it would not affect the equations of motion, since μ ∇ μn = 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).

An unconstrained variation of 2 Λ (n ) is with respect to μ n and the metric gμν, and allows the four components of nμ to be varied independently. It takes the form

ν 1-μ ν δΛ = μνδn + 2n μ δgμν, (146 )
∂ Λ μν = ℬn ν, ℬ ≡ − 2∂n2-. (147 )
The use of the letter ℬ is to remind us that this is a bulk fluid effect, which is present regardless of the number of fluids and constituents. The momentum covector μ ν is what we encountered earlier. It is dynamically, and thermodynamically, conjugate to nμ, and its magnitude can be seen to be the chemical potential of the particles by recalling that Λ = − ρ. Correctly identifying the momentum is essential for a concise and transparent formulation of fluids, e.g. for constructing proofs about vorticity, and also for coupling in dissipative terms in the equations of motion (cf. the discussion of Section 7). For later convenience, we also introduce the momentum form defined as
μμνλ = εμνλρμρ. (148 )

If the variation of the four-current was left unconstrained, the equations of motion for the fluid deduced from the above variation of Λ would require, incorrectly, that the momentum covector μμ should vanish in all cases. This reflects the fact that the variation of the conserved four-current must be constrained, meaning that not all components of μ n can be treated as independent. In terms of the constrained Lagrangian displacement of Equation (132View Equation), a first-order variation of the general relativity plus fluid Lagrangian yields

(√ --- [ 1 ]) √ ---[ 1 1 ] √ --- δ − g ----R + Λ = − g − ----G μν + --(Ψδμλ + n μμλ)gλν δgμν − − gf νξν 16π 16 π 2 +boundary terms, (149 )
where the boundary terms do not contribute to the field equations or stress-energy-momentum tensor (the divergence theorem implies that they become boundary terms in the action), the force density fν is defined in Equation (102View Equation), and the generalized pressure Ψ is defined to be
Ψ = Λ − nμμ . (150 ) μ
We have utilized the well-known formula [80] for the variation of the metric determinant, which can be found in Equation (370View Equation) of Appendix A. We also note that the fluid boundary term (FBT) is
( 1√ --- νλτ μ) FBT = ∇ ν -- − gμ nλτμξ , (151 ) 2
so that μνλτnλτμξμ on the boundary is fixed (i.e. ξμ vanishes) to have a well-posed action principle.

At this point we can return to the view that nμ is the fundamental field for the fluid. The principle of least action implies that the coefficients of ν ξ and δg μν and the boundary terms must vanish. Thus, the equations of motion consist of the original conservation condition from Equation (121View Equation), the Euler equation

μ fν = n ωμν = 0, (152 )
and of course the Einstein equation. When δgμν ⁄= 0, it is well-established in the relativity literature that the stress-energy-momentum tensor follows from a variation of the Lagrangian with respect to the metric, that is
μν --2-- (√ --- ) μ μ λν T δgμν ≡ √ −-gδ − gℒ = (Ψ δ λ + n μ λ)g δgμν. (153 )
When the complete set of field equations is satisfied then we see from Equation (105View Equation), which applies here, that it is automatically true that ∇ μT μν = 0.

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 (120View Equation)). There is thus much purpose in using the particular symbol fν in Equation (152View Equation). We may take the fν to be the relativistic analogue of the left-hand-side of Equation (120View Equation) 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 (152View Equation).

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