### 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 [660].)

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.