- Observe that the signature of this effective metric is indeed , as it should be to be regarded as Lorentzian.
- Observe that in physical acoustics it is the inverse metric density, that is of more fundamental significance for deriving the wave equation than is the metric itself. (This observation continues to hold in more general situations where it is often significantly easier to calculate the tensor density than it is to calculate the effective metric .)
- It should be emphasised that there are two distinct metrics relevant to the current discussion:
- The physical spacetime metric is in this case just the usual flat metric of Minkowski space: (Here is the speed of light in vacuum.) The fluid particles couple only to the physical metric . In fact the fluid motion is completely non-relativistic, so that , and it is quite sufficient to consider Galilean relativity for the underlying fluid mechanics.
- Sound waves on the other hand, do not “see” the physical metric at all. Acoustic perturbations couple only to the effective acoustic metric .

- It is quite remarkable that (to the best of our knowledge) the acoustic metric was first derived and used in Moncrief’s studies of the relativistic hydrodynamics of accretion flows surrounding black holes [268]. Indeed Moncrief was working in the more general case of a curved background “physical” metric, in addition to a curved “effective” metric. We shall come back to this work later on, in our historical section.
- The geometry determined by the acoustic metric does however inherit some key properties from the existence of the underlying flat physical metric. For instance, the topology of the manifold does not depend on the particular metric considered. The acoustic geometry inherits the underlying topology of the physical metric - ordinary - with possibly a few regions excised (due to whatever hard-wall boundary conditions one might wish to impose on the fluid). In systems constrained to have effectively less than 3 spacelike dimensions one can reproduce more complicated topologies (consider for example an effectively one-dimensional flow in a tubular ring).
- Furthermore, the acoustic geometry automatically inherits from the underlying Newtonian time parameter, the important property of “stable causality” [164, 422]. Note that This precludes some of the more entertaining causality-related pathologies that sometimes arise in general relativity. (For a general discussion of causal pathologies in general relativity, see for example [164, 161, 162, 72, 163, 396]).
- Other concepts that translate immediately are those of “ergo-region”, “trapped surface”, “apparent horizon”, and “event horizon”. These notions will be developed more fully in the following subsection.
- The properly normalised four-velocity of the fluid is so that This four-velocity is related to the gradient of the natural time parameter by Thus the integral curves of the fluid velocity field are orthogonal (in the Lorentzian metric) to the constant time surfaces. The acoustic proper time along the fluid flow lines (streamlines) is and the integral curves are geodesics of the acoustic metric if and only if is position independent.
- Observe that in a completely general (3+1)-dimensional Lorentzian geometry the metric has 6 degrees
of freedom per point in spacetime. ( symmetric matrix independent components;
then subtract coordinate conditions).
In contrast, the acoustic metric is more constrained. Being specified completely by the three scalars , , and , the acoustic metric has at most degrees of freedom per point in spacetime. The equation of continuity actually reduces this to degrees of freedom, which can be taken to be and .

Thus the simple acoustic metric of this section can at best reproduce some subset of the generic metrics of interest in general relativity.

- A point of notation: Where the general relativist uses the word “stationary” the fluid dynamicist uses the phrase “steady flow”. The general-relativistic word “static” translates to a rather messy constraint on the fluid flow (to be discussed more fully below).
- Finally, we should emphasise that in Einstein gravity the spacetime metric is related to the distribution of matter by the non-linear Einstein-Hilbert differential equations. In contrast, in the present context, the acoustic metric is related to the distribution of matter in a simple algebraic fashion.

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