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10 The “Pull-Back” Formalism for Two Fluids

Having discussed the single fluid model, and how one accounts for stratification, it is time to move on to the problem of modeling 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.

In keeping with the previous Section 9, we will rely on use of the constituent index, which for all remaining formulas of this section will range over x,y,z = n,s. In the example that we consider the two fluids represent the particles (n) and the entropy (s). Once again, the number density four-currents, to be denoted μ n x, are taken to be separately conserved, meaning

∇ μnμx = 0. (169 )
As before, we use the dual formulation, i.e. introduce the three-forms
x μ μ 1 μνλτ x nνλτ = ενλτμn x, n x = 3!ε nνλτ. (170 )
Also like before, the conservation rules are equivalent to the individual three-forms being closed, i.e. 
∇ [μnxνλτ] = 0. (171 )
However, we need a formulation whereby such conservation obtains automatically, at least in principle.

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 XA x, and we recall that A, B, C,etc.= 1,2,3. This is depicted in Figure 8View Image, 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 8View Image (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 x N 1, x N 2, etc. (cf. Figure 7View Image). For the same reason, we have also not labelled (as in Figure 7View Image) the “push-forwards” (represented by the arrows) from the matter spaces to spacetime.

View Image

Figure 8: The push-forward from a point in the th x-constituent’s three-dimensional matter space (on the left) to the corresponding “fluid-particle” worldline in spacetime (on the right). The points in matter space are labelled by the coordinates {X1x,X2x,X3x}, and the constituent index x = n,s. There exist as many matter spaces as there are dynamically independent fluids, which for this case means two.

By pulling-back each constituent’s three-form onto its respective matter space we can once again construct three-forms that are automatically closed on spacetime, i.e. let

x x A B C nνλτ = N ABC∇ νX x ∇ λX x ∇ τX x , (172 )
where NAxBC is completely antisymmetric in its indices and is a function of the XAx. Using the same reasoning as in the single fluid case, the construction produces three-forms that are automatically closed, i.e. they satisfy Equation (171View Equation) identically. If we let the scalar fields A X x (as functions on spacetime) be the fundamental variables, they yield a representation for each particle number density current that is automatically conserved. The variations of the three-forms can now be derived by varying them with respect to the XA x.

Lagrangian displacements on spacetime for each fluid, to be denoted μ ξx, will now be introduced. They are related to the variations A δX x via

( ) δXA = − ∇ μXA ξμ. (173 ) x x x
It should be clear that the analogues of Equations (131View Equation, 132View Equation, 134View Equation, 135View Equation) for this two-fluid case are given by the same formulas except that each displacement and four-current will now be associated with a constituent index, using the decomposition
μ μ x μ n x = nxux , uμu x = − 1, (174 )
where ux = g uν μ μν x.

Associated with each constituent’s Lagrangian displacement is its own Lagrangian variation. These are naturally defined to be

Δx ≡ δ + ℒξx, (175 )
so that it follows that
x Δxn μλτ = 0, (176 )
as expected for the pull-back construction. Likewise, two-fluid analogues of Equations (138View Equation, 139View Equation, 140View Equation) exist which take the same form except that the constituent index is attached. However, in contrast to the ordinary fluid case, there are many more options to consider. For instance, we could also look at the Lagrangian variation of the first constituent with respect to the second constituent’s flow, i.e. Δsnn, or the other way around, i.e. Δnns. The Newtonian analogues of these Lagrangian displacements were essential to an analysis of instabilities in rotating superfluid neutron stars [7].

We are now in a position to construct an action principle that will yield the equations of motion and the stress-energy tensor. Again, the central quantity is the “master” function Λ, which is now a function of all the different scalars that can be formed from the n μx, i.e. the scalars nx together with

n2xy = n2yx = − gμνnμxn νy. (177 )
In the limit where all the currents are parallel, i.e. the fluids are comoving, − Λ corresponds again to the local thermodynamic energy density. In the action principle, Λ is the Lagrangian density for the fluids. It should be noted that our choice to use only the fluid currents to form scalars implies that the system is “locally isotropic” in the sense that there are no a priori preferred directions, i.e. the fluids are equally free to move in any direction. Structures like the crust believed to exist near the surface of a neutron star generally could be locally anisotropic, e.g. sound waves in the lattice move in the preferred directions associated with the lattice.

An unconstrained variation of Λ with respect to the independent vectors nμ x and the metric gμν takes the form

( ) ∑ 1 ∑ δΛ = μxν δnνx + --gλν( nμxμxλ) δgμν, (178 ) x={n,s} 2 x={n,s}
( ) μxν = gνμ ℬxn μx + 𝒜xyn μy , (179 ) 𝒜xy = 𝒜yx = − -∂-Λ-, for x ⁄= y. (180 ) ∂n2xy
The momentum covectors x μν are each dynamically, and thermodynamically, conjugate to their respective number density currents nμx, and their magnitudes are the chemical potentials. We see that 𝒜xy represents the fact that each fluid momentum μxν may, in general, be given by a linear combination of the individual currents n μ x. That is, the current and momentum for a particular fluid do not have to be parallel. This is known as the entrainment effect. We have chosen to represent it by the letter 𝒜 for historical reasons. When Carter first developed his formalism he opted for this notation, referring to the “anomaly” of having misaligned currents and momenta. It has since been realized that the entrainment is a key feature of most multi-fluid systems and it would, in fact, be anomalous to leave it out!

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 n2xy (x ⁄= y) because then we obviously have 𝒜xy = 0. Entrainment is an observable effect in laboratory superfluids [94Jump To The Next Citation Point110Jump To The Next Citation Point] (e.g. via flow modifications in superfluid 4He and mixtures of superfluid 3 He and 4 He). In the case of neutron stars, entrainment is an essential ingredient of the current best explanation for the so-called glitches [9597]. Carter [19Jump To The Next Citation Point] 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.

In terms of the constrained Lagrangian displacements, a variation of Λ now yields

( ) ( ) (√ --- ) 1√ --- ∑ √ ---∑ 1√ --- ∑ δ − gΛ = -- − g( Ψδ μλ + n μxμxλ) gλνδgμν − − g fxνξνx + ∇ ν( -- − g μ νxλτnxλτμξμx)(1,81 ) 2 x= {n,s} x={n,s} 2 x={n,s}
where x fν is as defined in Equation (164View Equation) except that the individual velocities are no longer parallel. The generalized pressure Ψ is now
∑ Ψ = Λ − n νxμxν. (182 ) x= {n,s}
At this point we will return to the view that μ n n and μ ns are the fundamental variables for the fluids. Because the ξμx are independent variations, the equations of motion consist of the two original conservation conditions of Equation (169View Equation), plus two Euler type equations
x fν = 0, (183 )
and of course the Einstein equation (obtained exactly as before by adding in the Einstein–Hilbert term). We also find that the stress-energy tensor is
μ μ ∑ μ x T ν = Ψ δ ν + nxμ ν. (184 ) x={n,s}
When the complete set of field equations is satisfied then it is automatically true that ∇ μTμν = 0. One can also verify that T μν is symmetric. The momentum form μ μνλ x entering the boundary term is the natural extension of Equation (159View Equation) to this two-fluid case.

It must be noted that Equation (183View Equation) is significantly different from the multi-constituent version of Equation (166View Equation). 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 (183View Equation) still represents two independent equations. If one takes entropy as an independent fluid, then the static and spherically symmetric solutions will exhibit thermal equilibrium [38Jump To The Next Citation Point]. 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.

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