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11.2 Two-constituent, single fluid case

Now consider the case when there are the two constituents with densities nn and ns, two conserved density currents nμn and n μs, two chemical potential covectors μnμ and μsμ, but still only one four-velocity u μ. The master function Λ depends on both n2 n and n2 s meaning that
x x ν xy ν δμμ = ℬ μνδnx + 𝒳 μνδny, (193 )
where
xy ∂ℬx- σ ρ 𝒳μν = − 2 ∂n2 gμσgνρnxny (194 ) y
and ℬsμν is as in Equation (190View Equation) except that each n is replaced with s. We have chosen to use the letter 𝒳 to represent what is a true multi-constituent effect, which arises due to composition gradients in the system. An alternative would have been to use 𝒞 since the effect is due to the presence of different constituents. However, in his papers Carter tends to use ℬs = 𝒞, referring to the bulk entropy contribution as “caloric”. Our chosen notation is intended to avoid confusion. It is also the case that the presence of the composition term 𝒳 xμyν has not been emphasized in previous work. This may be unfortunate since a composition variation is known to affect the dynamics of a system, e.g. by giving rise to the g-modes in a neutron star [98].

The fact that n μ s is parallel to nμ n implies that it is only the magnitude of the entropy density current that is independent. One can show that the condition of transverse propagation, as applied to both currents, implies

δnμs = xsδn μn. (195 )
Now, we proceed as in the previous example, noting that the equation of motion is n s δfμ + δfμ = 0, which reduces to
( [ ] 2 [ ]) ℬn + x2ℬs -k0--− ℬnc2 + x2ℬsc2 − 2x 𝒳 ns kδni = 0, (196 ) s kjkj n s s s 00 i n
where
x c2x ≡ 1 + ∂-ln-ℬ--. (197 ) ∂ ln nx
The speed of sound is thus
2 ℬnc2n + x2sℬsc2s − 2xs 𝒳n0s0 C S = --------n----2--s------. (198 ) ℬ + xsℬ

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