### 7.3 Nontrivial dispersion as Einstein-aether theory

There is a certain precise sense in which nontrivial dispersion relations can effectively be viewed as
implicitly introducing an “aether field”, in the sense of providing a kinematic (but not dynamic)
implementation of Einstein-aether theory [323, 180, 219, 314]. The point is that to define
nontrivial dispersion one needs to pick a rest frame , and then assert in this rest
frame. But one can then re-write this dispersion relation (in the eikonal approximation) as
That is, using ,
As long as the background is slowly varying, this can be re-written as:
with and with the aether field hiding in the definition of the spatial Laplacian
. This procedure allows us to take a quantity that is manifestly not Lorentz
invariant, the dispersion relation , and nevertheless “covariantise” it via the introduction of
new structure — a locally specified preferred frame defined by the (possibly position- and time- dependent)
aether 4-velocity .
Of course, in standard analogue models such an aether field does not come with its own dynamics: It is a
background structure which breaks the physically-relevant content of what is usually called
diffeomorphism invariance (see next Section 7.4). However, in a gravitation theory context one
might still want to require background independence taking it as a fundamental property of
any gravity theory, even a Lorentz breaking one. In this case one has to provide the aether
field with a suitable dynamics; we can then rephrase much of the analogue gravity discussion
in the presence of nontrivial dispersion relations in terms of a variant of the Einstein-aether
models [323, 180, 219, 314].