Theorem. If a fluid is barotropic and inviscid, and the flow is irrotational (though possibly time dependent) then the equation of motion for the velocity potential describing an acoustic disturbance is identical to the d’Alembertian equation of motion for a minimally coupled massless scalar field propagating in a (3+1)-dimensional Lorentzian geometry
Under these conditions, the propagation of sound is governed by an acoustic metric -Comment. It is quite remarkable that even though the underlying fluid dynamics is Newtonian, nonrelativistic, and takes place in flat space plus time, the fluctuations (sound waves) are governed by a curved (3+1)-dimensional Lorentzian (pseudo-Riemannian) spacetime geometry. For practitioners of general relativity this observation describes a very simple and concrete physical model for certain classes of Lorentzian spacetimes, including (as we shall later see) black holes. On the other hand, this discussion is also potentially of interest to practitioners of continuum mechanics and fluid dynamics in that it provides a simple concrete introduction to Lorentzian differential geometric techniques.
Proof. The fundamental equations of fluid dynamics [219, 221
, 264
, 353
] are the equation of continuity
Since this is a subtle issue that we have seen cause considerable confusion in the past, let us be even more explicit by asking the rhetorical question: “How can we tell the difference between a wind gust and a sound wave?” The answer is that the difference is to some extent a matter of convention - sufficiently low-frequency long-wavelength disturbances (wind gusts) are conventionally lumped in with the average bulk motion. Higher-frequency, shorter-wavelength disturbances are conventionally described as acoustic disturbances. If you wish to be hyper-technical, we can introduce a high-pass filter function to define the bulk motion by suitably averaging the exact fluid motion. There are no deep physical principles at stake here - merely an issue of convention. The place where we are making a specific physical assumption that restricts the validity of our analysis is in the requirement that the amplitude of the high-frequency short-wavelength disturbances be small. This is the assumption underlying the linearization programme, and this is why sufficiently high-amplitude sound waves must be treated by direct solution of the full equations of fluid dynamics.
Linearizing the continuity equation results in the pair of equations
Now, the barotropic condition implies Use this result in linearizing the Euler equation. We obtain the pair This last equation may be rearranged to yield Use the barotropic assumption to relate Now substitute this consequence of the linearised Euler equation into the linearised equation of continuity. We finally obtain, up to an overall sign, the wave equation: This wave equation describes the propagation of the linearised scalar potentialNow, written in this form, the physical import of this wave equation is somewhat less than pellucid. To simplify things algebraically, observe that the local speed of sound is defined by
Now construct the symmetric Now in any Lorentzian (i.e., pseudo-Riemannian) manifold the curved space scalar d’Alembertian is
given in terms of the metric by (see, for example, [125, 266, 369, 265
, 164
, 422
])
We have presented the theorem and proof, which closely follows the discussion in [389], in considerable
detail because it is a standard template that can be readily generalised in many ways. This discussion
can then be used as a starting point to initiate the analysis of numerous and diverse physical
models.
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