Beyond these immediate questions, we could also seek similar effects in other physical or mathematical frameworks.
Adding external forces is easy, an early discussion can be found in  and more details are available in . The key point is that with an external force one can to some extent shape the background flow (see for example the discussion on ). Upon linearization, the fluctuations are however 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 symmetry. Similarly there is a real physical distinction between a truly (1+1)-dimensional system and a (3+1)-dimensional system with transverse symmetry. We emphasise that in cartesian coordinates the wave equationindependent 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 [376, 387, 391, 389].
Introducing the inverse acoustic metric , defined byspace (not spacetime). The covariant acoustic metric is then
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 super-fluid component of physical superfluids. (This point has been emphasised by Comer , 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.13 An approach similar to the spirit of the present discussion, but in terms of Clebsch potentials, can be found in . 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.)14
© Max Planck Society and the author(s)