7.2 Equivalence principle

Analogue models are of particular interest in that the analogue spacetimes that emerge often violate, to some extent, the Einstein equivalence principle [45Jump To The Next Citation Point, 638Jump To The Next Citation Point]. This is the heart and soul of any metric theory of gravity and is basically the requirement of the universality of free fall, plus local Lorentz invariance and local position invariance of non-gravitational experiments.

As such, the Einstein equivalence principle is a “principle of universality” for the geometrical structure of spacetime. Whatever the spacetime geometrical structure is, if all excitations “see” the same geometry, one is well on the way to satisfying the observational and experimental constraints. In a metric theory, this amounts to the demand of mono-metricity: A single universal metric must govern the propagation of all excitations.

Now it is this feature that is relatively difficult to arrange in analogue models. If one is dealing with a single degree of freedom, then mono-metricity is no great constraint. But with multiple degrees of freedom, analogue spacetimes generally lead to refringence – that is the occurrence of Fresnel equations that often imply multiple propagation speeds for distinct normal modes. To even obtain a bi-metric model (or, more generally, a multi-metric model), requires an algebraic constraint on the Fresnel equation that it completely factorises into a product of quadratics in frequency and wavenumber. Only if this algebraic constraint is satisfied can one assign a “metric” to each of the quadratic factors. If one further wishes to impose mono-metricity, then the Fresnel equation must be some integer power of some single quadratic expression, an even stronger algebraic statement [45, 638]. Faced with this situation, there are two ways in which the analogue gravity community might proceed:

  1. Try to find a broad class of analogue models (either physically based or mathematically idealised) that naturally lead to mono-metricity. Little work along these lines has yet been done; at least partially because it is not clear what features such a model should have in order to be “clean” and “compelling”.
  2. Accept refringence as a common feature of the analogue models and attempt to use refringence to ones benefit in one or more ways:

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