For example, given the machinery developed so far, taking the short wavelength/high frequency limit to obtain geometrical acoustics is now easy. Sound rays (phonons) follow the null geodesics of the acoustic metric. Compare this to general relativity where in the geometrical optics approximation light rays (photons) follow null geodesics of the physical spacetime metric. Since null geodesics are insensitive to any overall conformal factor in the metric [265, 164, 422] one might as well simplify life by considering a modified conformally related metric
We can rephrase this in a language more familiar to the acoustics community by invoking the Eikonal approximation. Express the linearised velocity potential, , in terms of an amplitude, , and phase, , by . Then, neglecting variations in the amplitude , the wave equation reduces to the Eikonal equation
As a sanity check on the formalism, it is useful to re-derive some standard results. For example, let the null geodesic be parameterised by . Then the null condition implies
Furthermore, if the geometry is stationary one can do slightly better. Let be some null path from to , parameterised in terms of physical arc length (i.e., ). Then the tangent vector to the path is.
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