### 6.2 Equivalence principle

Analogue models are of particular interest in that the analogue spacetimes that emerge often violate, to some extent, the Einstein equivalence principle [16399]. Since the Einstein equivalence principle (or more precisely the universality of free fall) is experimentally tested to the accuracy of about 1 part in , it is important to build this principle into realistic models of gravitation - most likely at a fundamental level.

One way of interpreting the Einstein equivalence principle is as 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 [16399].

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:
• There are real physical phenomena in non-gravitational settings that definitely do exhibit refringence and sometimes multi-metricity. Though situations of this type are not directly relevant to the gravity community, there is significant hope that the mathematical and geometrical tools used by the general relativity community might in these situations shed light on other branches of physics.
• Use the refringence that occurs in many analogue models as a way of “breaking” the Einstein equivalence principle, and indeed as a way of “breaking” even more fundamental symmetries and features of standard general relativity, with a view to exploring possible extensions of general relativity. While the analogue models are not themselves primary physics, they can nevertheless be used as a way of providing hints as to how more fundamental physics might work.