3.1 Historical period

Of course, the division into pre-1981 and post-1981 articles is at a deeper level somewhat deceptive. There have been several analogue models investigated over the years, with different aims, different levels of sophistication, and ultimately different levels of development. Armed with a good library and some hindsight it is possible to find interesting analogues in a number of places.16

3.1.1 Optics – the Gordon metric

Perhaps the first paper to seriously discuss analogue models and effective metric techniques was that of Gordon (yes, he of the Klein–Gordon equation) [258]. 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

[ −2] [geffective]μν = ημν + 1 − n V μVν, (139 )
where η μν is the flat Minkowski metric, n is the position-dependent refractive index, and V μ is the (constant) 4-velocity of the medium.

After that, there was sporadic interest in effective metric techniques. An historically-important contribution was one of the problems in the well-known book “The Classical Theory of Fields” by Landau and Lifshitz [373]. 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 [503], who showed that (under certain conditions) Maxwell’s equations can be expressed directly in terms of the effective metric specified by the coefficients

[ ] [geffective]μν = gμν + 1 − -1- V μVν, (140 ) 𝜖μ
where gμν is the ordinary spacetime metric, 𝜖 and μ are the permeability and permittivity, and Vμ 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 [576], Balazs [18], and Winterberg [689]. The general formalism was more fully developed in articles such as those by Plebański [511, 510], and a good summary of this classical period can be found in the article by de Felice [168]. 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

[magnetic permitivity] ∝ [electric permeability]. (141 )
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.

Subsequently, Anderson and Spiegel [7] extended and modified the notion of the Gordon metric to allow the medium to be a flowing fluid – so the 4-vector a V is no longer assumed to be a constant. In hindsight, using modern notation and working in full generality, the Gordon metric can be justified as follows: Consider an arbitrary curved spacetime with physical metric gab containing a fluid of 4-velocity V a and refractive index n. Pick a point in the manifold and adopt Gaussian normal coordinates around that point so that gab → ηab. Now perform a Lorentz transformation to go to to a local inertial frame co-moving with the fluid, so that a ⃗ V → (1;0 ) and one is now in Gaussian normal coordinates co-moving with the fluid. In this coordinate patch, the light rays, by definition of the refractive index, locally propagate along the light cones

-1- 2 2 − n2 dt + ||d⃗x|| = 0, (142 )
implying in these special coordinates the existence of an optical metric
⌊ ⌋ − 1∕n2 0 0 0 | 0 1 0 0 | 𝒢μν ∝ |⌈ 0 0 1 0 |⌉ . (143 ) 0 0 0 1
That is, transforming back to arbitrary coordinates:
{ } 1 1 𝒢 μν ∝ − n2VμV ν + {gμν + V μVν} ∝ gμν + 1 − n2- VμVν. (144 )
Note that in the ray optics limit, because one only has the light cones to work with, one can neither derive nor is it even meaningful to specify the overall conformal factor.

3.1.2 Acoustics

After the pioneering hydrodynamical paper by White in 1973 [687], which studied acoustic ray tracing in non-relativistic moving fluids, there were several papers in the 1980s using an acoustic analogy to investigate the propagation of shockwaves in astrophysical situations, most notably those of Moncrief [448Jump To The Next Citation Point] and Matarrese [433Jump To The Next Citation Point, 434Jump To The Next Citation Point, 432Jump To The Next Citation Point]. In particular, in Moncrief’s work [448Jump To The Next Citation Point] the linear perturbations of a relativistic perfect fluid on a Schwarzschild background 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 [448Jump To The Next Citation Point] can be seen as a precursor to the later works on acoustic geometries and acoustic horizons. Indeed, because they additionally permit a general relativistic Schwarzschild background, the results of Moncrief [448Jump To The Next Citation Point] are, in some sense, more general than those considered in the mainstream acoustic gravity papers that followed.

In spite of these impressive results, we consider these papers to be part of the “historical period” for the main reason that such works are philosophically orthogonal to modern developments in analogue gravity. Indeed the main motivation for such works was the study of perfect fluid dynamics in accretion flows around black holes, or in cosmological expansion, and in this context the description via an acoustic effective background was just a tool in order to derive results concerning conservation laws and stability. This is probably why, even if temporally, [448Jump To The Next Citation Point] pre-dates Unruh’s 1981 paper by one year, and while [433, 434, 432] post-date Unruh’s 1981 paper by a few years, there seems to have not been any cross-connection.

3.1.3 Surface waves

Somewhat ironically, 1983 marked the appearance of some purely experimental results on surface waves in water obtained by Badulin et al. [17Jump To The Next Citation Point]. At the time these results passed unremarked by the relativity community, but they are now of increasing interest, and are seen to be precursors of the theoretical work reported in [560Jump To The Next Citation Point, 531Jump To The Next Citation Point] and the modern experimental work reported in [532Jump To The Next Citation Point, 682Jump To The Next Citation Point].


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