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9 The Two-Constituent, Single Fluid

Generally speaking, the total energy density ρ can be a function of independent parameters other than the particle number density nn, such as the entropy density ns, assuming that the system scales in the manner discussed in Section 5 so that only densities need enter the equation of state. (For later convenience we will introduce the constituent indices x, y, etc. which range over the two constituents {n,s}, and do not satisfy any kind of summation convention.) If there is no heat conduction, then this is still a single fluid problem, meaning that there is still just one unit flow velocity uμ [36]. This is what we mean by a two-constituent, single fluid. We assume that the particle number and entropy are both conserved along the flow, in the same sense as in Equation (126View Equation). Associated with each parameter there is thus a conserved current density four-vector, i.e. μ μ n n = nnu for the particle number density and n μs = nsuμ for the entropy density. Note that the ratio xs = ns∕nn is comoving in the sense that

uμ∇ μ (xs) = 0. (154 )

In terms of constituent indices x, y, the associated combined first and second laws can be written in the form

dρ = ∑ μx dn = − ∑ μx dnν, (155 ) x ν x x={n,s} x={n,s}
since ρ = ρ(n2n,n2s), where
n μx = nxuμ, n2x = − gμνnμxn νx, (156 )
∂ ρ μxν = gνμℬxn μx, ℬx ≡ 2 --2. (157 ) ∂nx
Because of Equation (83View Equation) μs = T, where T is the temperature.

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, Ns, that can be attached to each worldline, like the particle number Nn of Figure 7View Image. In order to describe the worldlines we can use the same three scalars XA (xμ) 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 A i X (0,x ) scalars. Each worldline will have also the conserved numbers Nn and Ns assigned to them. Thus, the values of these numbers can be expressed as functions of the XA (0, xi). But most importantly, the fact that each Nx is conserved, means that this function specification must hold for all of spacetime, so that in particular the ratio xs is of the form μ A μ xs(x ) = xs(X (x )). Consequently, we now have a construction whereby this ratio identically satisfies Equation (154View Equation), and the action principle remains a variational problem just in terms of the three XA scalars.

The variation of the action follows like before, except now a constituent index x must be attached to the particle number density current and three-form:

nxνλτ = ενλτμn μx. (158 )
Once again it is convenient to introduce the momentum form, now defined as
μνλ μνλρ x μx = ε μρ. (159 )
Since the XA are the same for each nx νλτ, the above discussion indicates that the pull-back construction now is to be based on
x x ( A) ( B) C nνλτ = N ABC ∇ νX ∇ λX ∇ τX , (160 )
where x NABC is completely antisymmetric and a function of the A X. After some due deliberation, the reader should be convinced that the only thing required here in addition to the previous Section 8 is to attach an index x to n νλτ, nμ, and n in Equations (131View Equation), (132View Equation), and (134View Equation), respectively. If we now define the master function as
Λ = − ρ (161 )
and the generalized pressure Ψ to be
∑ x ν ∑ x Ψ = Λ − μνnx = Λ + μ nx, (162 ) x={n,s} x={n,s}
then the first-order variation for Λ is
( ) (√ --- ) 1√ --- √--- ∑ δ − gΛ = -- − g (Ψg μν + [Ψ − Λ ]uμuν) δgμν − − g ( f xν) ξν + 2 x= {n,s} ( ) ∇ ( 1√ −-g ∑ μ νλτnx ξμ) , (163 ) ν 2 x λτμ x= {n,s}
fxν = n μxωxμν, (164 )
ωxμν = 2∇ [μμxν]. (165 )
Keeping in mind that we need the Einstein–Hilbert action for the Einstein equation, the final fluid equations of motion are
∑ fx = 0, (166 ) ν x={n,s}
∇ νnνx = 0, (167 )
whereas the stress-energy-momentum tensor takes the familiar form
μν μν μ ν T = Ψg + (Ψ − Λ )u u . (168 )

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