A metric describes the apparent geometry of a region when probed using signals or test bodies – an acoustic metric describes a region in which the presence and properties of those signals is nonlinear, and physically changes the geometry.
In acoustics, an acoustic wave is caused by the transmission of density-waves in a particulate medium. This requires particles to shift their positions and briefly have forward-and-backward velocities as the wave passes through their location. If we imagine that a "loud" low-frequency wave is "rocking" the atoms back and forth in a region, and we then send a higher-frequency wave through the same region in a different direction, the waveshapes do not simply superimpose "additively" in a linear manner - the high-frequency signal experiences the low-frequency signal as a series of regions in which the medium is moving, and this makes the second signal transmit faster or slower through the zones as a result of the different pressures and different averaged background velocities of the medium in those zones – the initial low-frequency signal is not simply propagating through the metric, it is also modifying the shape of the metric, in such a way that the map of apparent distances and geometrical properties of the region when a signal is present is no longer the same as the map for when the region was "quiet".
This non-linear behaviour also appears even if we only have a single signal - producing the apparently paradoxical result that our map of how signals propagate through a region is only technically correct if no such signals are present(!). In practice, the linear approach is usually "good enough" to describe signals that are of reasonably low amplitude - if we clap our hands once, the shockwave will travel at the standard speed of sound [math]s[/math] calculated from the normal density and composition of the air. However, if we create a more extreme shockwave using high explosives, the wave can propagate at more than the background speed of sound, as the extreme physics taking place at the shockwave invalidates the usual assumptions that we used to calculate [math]s[/math] (air molecules being flung forwards at more than [math]s[/math], condensation of water vapor behind the shockwave, and so on). If we try to use a linear approach in these situation, we can appear to get apparently nonsensical results, with signals propagating at speeds other than the "official" propagation speed of the medium.
Examples of nonlinearity
- When an extremely high-energy EM wave passes through a region, energy-densities may be high enough for the creation of particle-pairs, which ... even if the pairs then self-annihilate almost immediately ... can affect how the signal propagates.
- Gravitational waves' – gravity-waves are generally assumed to travel at the same speed as light, [math]c[/math]. But since a variation in gravitational properties changes the speed of light, how fast should a gravity-wave propagate in practice? (see: the gravity-wave problem).
- In gravitational theory, we might also consider a moving gravitational mass to count as type of gravitational signal (as it carries a gravitational payload from place to place). Given that he speed of the body is a function of it's momentum and energy divided by its inertia, along with the background properties of the region that it passes through, then if its presence changes those background properties, how fast does the object move?