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14.3 Carter’s canonical framework

The most recent attempt to construct a relativistic formalism for dissipative fluids is due to Carter [20Jump To The Next Citation Point]. His approach is less “phenomenological” than the ones we have considered so far in that it is based on making maximal use of variational principle arguments. The construction is also extremely general. On the one hand this makes it more complex. On the other hand this generality could prove useful in more complicated cases, e.g. for investigations of multi-fluid dynamics and/or elastic media. Given the potential that this formalism has for future applications, it is worth discussing it in detail.

The overall aim is to generalize the variational formulation described in Section 8 in such a way that viscous “stresses” are accounted for. Because the variational foundations are the same, the number currents n μx play a central role (x is a constituent index as before). In addition we can introduce a number of viscosity tensors τμΣν, which we assume to be symmetric (even though it is clear that such an assumption is not generally correct [6]). The index Σ is “analogous” to the constituent index, and represents different viscosity contributions. It is introduced in recognition of the fact that it may be advantageous to consider different kinds of viscosity, e.g. bulk and shear viscosity, separately. As in the case of the constituent index, a repeated index Σ does not imply summation.

The key quantity in the variational framework is the master function Λ. As it is a function of all the available fields, we will now have μ μν Λ(nx,τΣ ,gμν). Then a general variation leads to

∑ x ρ 1-∑ Σ μν -∂Λ-- μν δΛ = x μ ρ δnx + 2 πμν δτΣ + ∂g μν δg . (308 ) Σ
Since the metric piece is treated in the same way as in the non-dissipative problem we will concentrate on the case δgμν = 0. This is, obviously, no real restriction since we already know how the variational principle couples in the Einstein equations in the general case. In the above formula we recognize the momenta x μρ that are conjugate to the fluxes. We also have an analogous set of “strain” variables which are defined by
∂Λ πΣμν = πΣ(μν) = 2--μν-. (309 ) ∂τΣ

As in the non-dissipative case, the variational framework suggests that the equations of motion can be written as a force-balance equation,

∑ ∑ ∇ μTνμ= fxν + fΣν = 0, (310 ) x Σ
where the generalized forces work out to be
x x σ σ x fρ = μ ρ∇ σn x + nx∇ [σμ ρ] (311 )
( ) f Σ= πΣ ∇ τσν + τσν ∇ πΣ − 1∇ πΣ . (312 ) ρ ρν σ Σ Σ σ ρν 2 ρ σν
Finally, the stress-energy tensor becomes
μ μ ∑ μ ∑ μσ Σ T ν = Ψg ν + μνnx + τΣ πσν, (313 ) x Σ
with the generalized pressure given by
Ψ = Λ − ∑ μx nρ − 1-∑ τρσπΣ . (314 ) x ρ x 2 Σ Σ ρσ

For reasons that will become clear shortly, it is useful to introduce a set of “convection vectors”. In the case of the currents, these are taken as proportional to the fluxes. This means that we introduce β μ x according to

μ μ x ν x ν hxβx = nx, μ νβx = − 1 − → hx = − μνn x. (315 )
With this definition we can introduce a projection operator
⊥ μν= gμν + μμβ ν −→ ⊥ μ βν = ⊥ μνμx = 0. (316 ) x x x xν x x ν
From the definition of the force density f μx we can now show that
∇μn μx = − βμxfμx (317 )
h ℒ μx = ⊥x f σ, (318 ) x x ν νσ x
where the Lie-derivative along β μx is ℒx = ℒβμx. We see that the component of the force which is parallel to the convection vector βμx is associated with particle conservation. Meanwhile, the orthogonal component represents the “change in momentum” along βμ x.

In order to facilitate a similar decomposition for the viscous stresses, it is natural to choose the conduction vector as a unit null eigenvector associated with μν πΣ. That is, we take

πΣ β ν= 0 (319 ) μν Σ
together with
Σ ν Σ μ u μ = gμνβΣ and uμ βΣ = − 1. (320 )
Introducing the projection associated with this conduction vector,
⊥ Σ = g + uΣu Σ, (321 ) μν μν μ ν
Σ ν ⊥ μν βΣ = 0. (322 )
Once we have defined βμ Σ, we can use it to reduce the degrees of freedom of the viscosity tensors. So far, we have only required them to be symmetric. However, in the standard case one would expect a viscous tensor to have only six degrees of freedom. To ensure that this is the case in the present framework we introduce the following degeneracy condition:
Σ μν uμτΣ = 0. (323 )
That is, we require the viscous tensor τμΣν to be purely spatial according to an observer moving along u μ Σ. With these definitions one can show that
β μℒ πΣ = 0, (324 ) Σ Σ μν
where ℒΣ = ℒβμ Σ is the Lie-derivative along β μΣ, and
τμνℒΣ πΣ = − 2βμf Σ. (325 ) Σ μν Σ μ

Finally, let us suppose that we choose to work in a given frame, moving with four-velocity u μ (and that the associated projection operator is ⊥ μ ν). Then we can use the following decompositions for the conduction vectors:

μ μ μ μ μ μ β x = βx(u + vx) and βΣ = β Σ(u + vΣ). (326 )
We see that μx = 1∕β x represents a chemical type potential for species x with respect to the chosen frame. At the same time, it is easy to see that Σ μ = 1 ∕βΣ is a Lorentz contraction factor. Using the norm of μ βΣ we have
βμβ Σ = − β2 (1 − v2) = − 1, (327 ) Σ μ Σ Σ
where v2 = vμv Σ Σ Σ μ. Thus
------- Σ ∘ 2 μ = 1∕ βΣ = 1 − vΣ, (328 )
which is clearly analogous to the standard Lorentz contraction formula.

So far the construction is quite formal, but we are now set to make contact with the physics. First we note that the above results allow us to demonstrate that

∑ ∑ ( ) uσ∇ ρT ρσ + (μx∇ σnσ + vσf x) + vσfΣ + 1μΣ τρσℒΣ πΣ = 0. (329 ) x x x σ Σ Σ σ 2 Σ ρσ
Recall that similar results were central to expressing the second law of thermodynamics in the previous two Sections 14.1 and 14.2. To see how things work out in the present formalism, let us single out the entropy fluid (with index s) by defining sμ = nμs and T = μs. To simplify the final expressions it is also useful to assume that the remaining species are governed by conservation laws of the form
μ ∇ μnx = Γ x, (330 )
subject to the constraint of total baryon conservation
∑ Γ x = 0. (331 ) x⁄=s
Given this, and the fact that the divergence of the stress-energy tensor should vanish, we have
μ ∑ x ∑ μ x ∑ ( μ Σ 1 Σ μν Σ ) T∇ μs = − μ Γ x − vxfμ − vΣfμ + 2μ τΣ ℒ Σπμν . (332 ) x⁄=s x Σ
Finally, we can bring the remaining two force contributions together by introducing the linear combinations
∑ ∑ ζxvμx = 0 and ζxΣv μx = vμΣ, (333 ) x x
constrained by
∑ ∑ ζx = ζxΣ = 1. (334 ) x x
Then defining
∑ f˜νx= fxν + ζxΣfΣν , (335 ) Σ
we have
μ ∑ x ∑ μ ˜x 1-∑ Σ μν Σ T∇ μs = − μ Γ x − vx fμ − 2 μ τΣ ℒ Σπ μν ≥ 0. (336 ) x⁄=s x Σ
The three terms in this expression represent, respectively, the entropy increase due to (i) chemical reactions, (ii) conductivity, and (iii) viscosity. The simplest way to ensure that the second law of thermodynamics is satisfied is to make each of the three terms positive definite.

At this point, the general formalism must be completed by some suitably simple model for the various terms. A reasonable starting point would be to assume that each term is linear. For the chemical reactions this would mean that we expand each Γ x according to

∑ Γ x = − 𝒞xyμy, (337 ) y⁄=s
where 𝒞xy is a positive definite (or indefinite) matrix composed of the various reaction rates. Similarly, for the conductivity term it is natural to consider “standard” resistivity such that
f˜x= − ∑ ℛxy vσ. (338 ) ρ y ρσ y
Finally, for the viscosity we can postulate a law of form
τμΣν = − ημΣνρσℒ ΣπΣρσ, (339 )
where we would have, for an isotropic model,
ημΣνρσ = ηΣ ⊥ μΣ(ρ⊥ σΣ)ν+ 1-(ηΣ − ζΣ) ⊥μΣν⊥ ρΣσ, (340 ) 3
where the coefficients ηΣ and ζΣ can be recognized as representing shear and bulk viscosity, respectively.

A detailed comparison between Carter’s formalism and the Israel–Stewart framework has been carried out by Priou [89Jump To The Next Citation Point]. He concludes that the two models, which are both members of a large family of dissipative models, have essentially the same degree of generality and that they are equivalent in the limit of linear perturbations away from a thermal equilibrium state. Providing explicit relations between the main parameters in the two descriptions, he also emphasizes the key point that analogous parameters may not have the same physical interpretation.

In developing his theoretical framework, Carter argued in favor of an “off the peg” model for heat conducting models [18]. This model is similar to that introduced in Section 10, and was intended as a simple, easier to use alternative to the Israel–Stewart construction. In the particular example discussed by Carter, he chooses to set the entrainment between particles and entropy to zero. This was done in order to simplify the discussion. But, as a discussion by Olson and Hiscock [87] shows, it has disastrous consequences. The resulting model violates causality in two simple model systems. As discussed by Priou [89] and Carter and Khalatnikov [25Jump To The Next Citation Point], this breakdown emphasizes the importance of the entrainment effect in these problems.

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