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11.1 Single fluid case

Suppose that there is only one constituent, with index x = n. The master function Λ then depends only on n2n. The variation in the chemical potential due to a small disturbance δnμn is
δμnμ = ℬnμνδnνn, (189 )
where
n n ∂-ℬn σ ρ ℬ μν = ℬ gμν − 2 ∂n2g μσgνρnnnn. (190 ) n
The equation of motion is δfμn= 0. It is not difficult to show, by using the condition of transverse wave propagation (188View Equation) and contracting with the spatial part of the wave vector ki = gijkj, that the equation of motion reduces to
( ) n n kjkj i ℬ + ℬ00--2-- kiδnn = 0. (191 ) k0
The speed of sound is thus
2 n 2 -k0- ∂-ln-ℬ-- CS ≡ kiki = 1 + ∂ ln nn . (192 )
Given that a well-constructed fluid model should have 2 C S ≤ 1, we see that the second term must be negative. This would ensure that the model is causal.
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