- Adding external forces.
- Working in truly (1+1) or (2+1) dimensional systems.
- Adding vorticity, to go beyond the irrotational constraint.

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

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

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

where 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 [607, 622, 626, 624].Introducing the inverse acoustic metric , defined by

the wave Equation (119) corresponds to the d’Alembertian wave equation in a curved space-time with contravariant metric tensor: where is the dimension of space (not spacetime). The covariant acoustic metric is then

- d = 3:
- The acoustic line element for three space and one time dimension reads
- d = 2:
- The acoustic line element for two space and one time dimension reads 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 continues to make good sense; it is only the step from 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 and above.

For the preceding analysis to hold, it is necessary and sufficient that the flow locally be vorticity free, , 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 [143], 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 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 [502]. 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}

Living Rev. Relativity 14, (2011), 3
http://www.livingreviews.org/lrr-2011-3 |
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