Perhaps the first paper to seriously discuss analogue models and effective metric techniques was that of Gordon (yes, he of the Klein-Gordon equation) [151]. Note that Gordon seemed largely interested in trying to describe dielectric media by an “effective metric”. That is: Gordon wanted to use a gravitational field to mimic a dielectric medium. What is now often referred to as the Gordon metric is the expression

where is the flat Minkowski metric, is the refractive index, and is the 4-velocity of the medium.After that, there was sporadic interest in effective metric techniques. One historically important contribution was one of the problems in the well-known book “The classical theory of fields” by Landau and Lifshitz [222]. See the end of chapter 10, paragraph 90, and the problem immediately thereafter: “Equations of electrodynamics in the presence of a gravitational field”. Note that in contrast to Gordon, here the interest is in using dielectric media to mimic a gravitational field.

In France the idea was taken up by Pham Mau Quan [308], who showed that (under certain conditions) Maxwell’s equations can be expressed directly in terms of the effective metric specified by the coefficients

where is the ordinary spacetime metric, and are the permeability and permittivity, and is the 4-velocity of the medium. The trajectories of the electromagnetic rays are interpreted in this case as geodesics of null length of this new effective metric.Three articles that directly used the dielectric analogy to analyse specific physics problems are those of Skrotskii [352], Balazs [8], and Winterberg [427]. The general formalism was more fully developed in articles such as those by Peblanski [304, 303], and good summary of this classical period can be found in the article by de Felice [100].

In summary and with the benefit of hindsight: An arbitrary gravitational field can always be represented as an equivalent optical medium, but subject to the somewhat unphysical restriction that

If an optical medium does not satisfy this constraint (with a position independent proportionality constant) then it is not completely equivalent to a gravitational field. For a position dependent proportionality constant complete equivalence can be established in the geometric optics limit, but for wave optics the equivalence is not complete.

There were several papers in the 1980’s using an acoustic analogy to investigate the propagation of shockwaves
in astrophysical situations, most notably those of Moncrief [268] and Matarrese [259, 260, 258]. In
particular in Moncrief’s work [268] the linear perturbations of a relativistic perfect fluid on an arbitrary
general relativistic metric were studied, and it was shown that the wave equation for such perturbations can
be expressed as a relativistic wave equation on some effective (acoustic) metric (which can admit acoustic
horizons). In this sense [268] can be seen as a precursor to the later works on acoustic geometries and acoustic
horizons.^{18}

The so-called “electro-mechanical analogy” has also had a long history within the engineering community. It
is sometimes extended to obtain an “electro-mechanical-acoustic” analogy, or even an “electro-thermal” analogy.
Unfortunately the issues of interest to the engineering community rarely resonate within the relativity community,
and these engineering analogies (though powerful in their own right) have no immediate impact for our
purposes.^{19}

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