7.4 Diffeomorphism invariance

When looking at the analogue metrics one problem immediately comes to mind. The laboratory in which the condensed-matter system is set up provides a privileged coordinate system. Thus, one is not really reproducing a geometrical configuration but only a specific metrical representation of it. This naturally raises the question of whether or not diffeomorphism invariance is lost in the analogue spacetime construction. Indeed, if all the degrees of freedom contained in the metric had a physical role, as opposed to what happens in a general relativistic context in which only the geometrical degrees of freedom (metric modulo diffeomorphism gauge) are physical, then diffeomorphism invariance would be violated. Here we are thinking of “active” diffeomorphisms, not “passive” diffeomorphisms (coordinate changes). As is well known, any theory can be made invariant under passive diffeomorphisms (coordinate changes) by adding a sufficient number of external/background/non-dynamical fields (prior structure). See, for instance, [256]. Invariance under active diffeomorphisms is equivalent to the assertion that there is no “prior geometry” (or that the prior geometry is undetectable). Many readers may prefer to re-phrase the current discussion in terms of the undetectability of prior structure.

The answer to this question is that active diffeomorphism invariance is maintained but only for (low-energy) internal observers, i.e., those observers who can only perform (low-energy) experiments involving the propagation of the relativistic collective fields. By revisiting classic Lorentz–FitzGerald ideas on length contraction, and analyzing the Michelson–Morley experiment in this context, it has been explicitly shown in [36] that (low-energy) Lorentz invariance is not broken, i.e., that an internal observer cannot detect his absolute state of motion. (For earlier suggestions along these lines, see, for example, [400] and [660Jump To The Next Citation Point].)

The argument is the same for curved spacetimes; the internal observer would have no way to detect the “absolute” or fixed background. So the apparent background dependence provided by the (non-relativistic) condensed-matter system will not violate active diffeomorphism invariance, at least not for these internal inhabitants. These internal observers will then have no way to collect any metric information beyond what is coded into the intrinsic geometry (i.e., they only get metric information up to a gauge or diffeomorphism equivalence factor). Internal observers would be able to write down diffeomorphism invariant Lagrangians for relativistic matter fields in a curved geometry. However, the dynamics of this geometry is a different issue. It is a well-known issue that the expected relativistic dynamics, i.e., the Einstein equations, have to date not been reproduced in any known condensed-matter system.

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