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 [323Jump To The Next Citation Point, 180Jump To The Next Citation Point, 219Jump To The Next Citation Point, 314Jump To The Next Citation Point]. The point is that to define nontrivial dispersion one needs to pick a rest frame V a, and then assert ω2 = f(k2) in this rest frame. But one can then re-write this dispersion relation (in the eikonal approximation) as
[ a 2 ab a b ] − (V ∂a) + f ([g + V V ]∂a∂b) Ψ (x) = 0. (313 )
That is, using f(w) = j(w ) + w,
[gab∂ ∂ + j([gab + V aV b]∂ ∂ )] Ψ (x ) = 0. (314 ) a b a b
As long as the background is slowly varying, this can be re-written as:
[Δd+1 + j(Δd )]Ψ (x) = 0, (315 )
with Δd+1 = gab∇a ∇b and with the aether field V a hiding in the definition of the spatial Laplacian Δ = [gab + V aV b]∇ ∇ d a b. This procedure allows us to take a quantity that is manifestly not Lorentz invariant, the dispersion relation 2 2 ω = f(k ), 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 V a.

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].

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