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11 Speeds of Sound

Crucial to the understanding of black holes is the effect of spacetime curvature on the light-cone structures, that is, the totality of null vectors that emanate from each spacetime point. Crucial to the propagation of massless fields is the light-cone structure. In the case of fluids, it is both the speed of light and the speed (and/or speeds) of sound that dictate how waves propagate through the matter, and thus how the matter itself propagates. We give here a simple analysis that uses plane-wave propagation to derive the speed(s) of sound in the single-fluid, two-constituent single-fluid, and two-fluid cases, assuming that the metric variations vanish (see [19Jump To The Next Citation Point] for a more rigorous derivation). The analysis is local, assuming that the speed of sound is a locally defined quantity, and performed using local, Minkowski coordinates x μ. The purpose of the analysis is to illuminate how the presence of various constituents and multi-fluids impact the local dynamics.

In a small region we will assume that the configuration of the matter with no waves present is locally isotropic, homogeneous, and static. Thus for the background n μx = {nx,0,0,0} and the vorticity ωxμν vanishes. For all cases the general form of the variation of the force density f x ν for each constituent is

δf x= 2nμ ∂ δμx . (185 ) ν x [μ ν]
Similarly, the conservation of the n μx gives
∂ δnμ = 0. (186 ) μ x
We are here using the point of view that the n μx are the fundamental fluid fields, and thus plane wave propagation means
δn μ= Aμ eikνxν, (187 ) x x
where the amplitudes A μx and the wave vector kμ are constant. Equations (186View Equation) and (187View Equation) imply simply
kμδnμ = 0, (188 ) x
i.e. the waves are “transverse” in the spacetime sense. We will give general forms for δμxν in what follows, and only afterwards assume static, homogeneous, and isotropic backgrounds.

 11.1 Single fluid case
 11.2 Two-constituent, single fluid case
 11.3 Two fluid case

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