2.7 Generalizing the physical model

There are a large number of ways in which the present particularly-simple analogue model can be generalised. Obvious issues within the current physical framework are:

Beyond these immediate questions, we could also seek similar effects in other physical or mathematical frameworks.

2.7.1 External forces

Adding external forces is (relatively) easy, an early discussion can be found in [624Jump To The Next Citation Point] and more details are available in [641Jump To The Next Citation Point]. The key point is that with an external force one can to some extent shape the background flow (see for example the discussion in [249Jump To The Next Citation Point]). However, upon linearization, the fluctuations are insensitive to any external force.

2.7.2 The role of dimension

The role of spacetime dimension in these acoustic geometries is sometimes a bit surprising and potentially confusing. This is important because there is a real physical distinction, for instance, between truly (2+1)-dimensional systems and effectively (2+1)-dimensional systems in the form of (3+1)-dimensional systems with cylindrical (pancake-like) symmetry. Similarly, there is a real physical distinction between a truly (1+1)-dimensional system and a (3+1)-dimensional system with transverse (cigar-like) symmetry. We emphasise that in Cartesian coordinates the wave equation

( ) --∂- f μν-∂--ϕ = 0, (119 ) ∂x μ ∂xν
[ − ρ∕c2 | − ρvj ∕c2 ] fμν = -----i-2-|--ij----ij--2-- , (120 ) − ρv ∕c ρ{δ − v v ∕c }
holds independent of the dimensionality of spacetime. It depends only on the Euler equation, the continuity equation, a barotropic equation of state, and the assumption of irrotational flow [607Jump To The Next Citation Point, 622Jump To The Next Citation Point, 626Jump To The Next Citation Point, 624Jump To The Next Citation Point].

Introducing the inverse acoustic metric μν g, defined by

√ --- 1 f μν = − gg μν; g = -------- , (121 ) det (g μν)
the wave Equation (119View Equation) corresponds to the d’Alembertian wave equation in a curved space-time with contravariant metric tensor:
( )−2∕(d− 1)[ − 1∕c2 | − vT ∕c2 ] g μν = ρ- --------|----------------- , (122 ) c − v∕c2 Id×d − v ⊗ vT ∕c2
where d is the dimension of space (not spacetime). The covariant acoustic metric is then
[ | ] (ρ-)2∕(d− 1) −-(c2 −-v2) −-vT- gμν = c − v Id×d . (123 )

d = 3:
The acoustic line element for three space and one time dimension reads
( ) [ 2 2 | T ] g = ρ- -−-(c-−-v-)-−-v-- . (124) μν c − v I3×3
d = 2:
The acoustic line element for two space and one time dimension reads
[ | ] ( ρ)2 −-(c2-−-v2) −-vT- gμν = c − v |I2×2 . (125)
This situation would be appropriate, for instance, when dealing with surface waves or excitations confined to a particular substrate.
d = 1:
The naive form of the acoustic metric in (1+1) dimensions is ill-defined, because the conformal factor is raised to a formally infinite power. This is a side effect of the well-known conformal invariance of the Laplacian in 2 dimensions. The wave equation in terms of the densitised inverse metric fμν continues to make good sense; it is only the step from f μν to the effective metric that breaks down. Acoustics in intrinsically (1+1) dimensional systems does not reproduce the conformally-invariant wave equation in (1+1) dimensions.

Note that this issue only presents a difficulty for physical systems that are intrinsically one-dimensional. A three-dimensional system with plane symmetry, or a two-dimensional system with line symmetry, provides a perfectly well-behaved model for (1+1) dimensions, as in the cases d = 3 and d = 2 above.

2.7.3 Adding vorticity

For the preceding analysis to hold, it is necessary and sufficient that the flow locally be vorticity free, ∇ × v = 0, so that velocity potentials exist on an atlas of open patches. Note that the irrotational condition is automatically satisfied for the superfluid component of physical superfluids. (This point has been emphasised by Comer [143Jump To The Next Citation Point], who has also pointed out that in superfluids there will be multiple acoustic metrics – and multiple acoustic horizons – corresponding to first and second sound.) Even for normal fluids, vorticity-free flows are common, especially in situations of high symmetry. Furthermore, the previous condition enables us to handle vortex filaments, where the vorticity is concentrated into a thin vortex core, provided we do not attempt to probe the vortex core itself. It is not necessary for the velocity potential ϕ to be globally defined.

Though physically important, dealing with situations of distributed vorticity is much more difficult, and the relevant wave equation is more complicated in that the velocity scalar is now insufficient to completely characterise the fluid flow.14 An approach similar to the spirit of the present discussion, but in terms of Clebsch potentials, can be found in [502Jump To The Next Citation Point]. The eikonal approximation (geometrical acoustics) leads to the same conformal class of metrics previously discussed, but in the realm of physical acoustics the wave equation is considerably more complicated than a simple d’Alembertian. (Roughly speaking, the vorticity becomes a source for the d’Alembertian, while the vorticity evolves in response to gradients in a generalised scalar potential. This seems to take us outside the realm of models of direct interest to the general relativity community.)15

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