How to understand the emergent special relativity in the superfluid?

The superfluid vacuum theory was proposed to understand some features of the vacuum (aether) from the emergence point of view. Although made up of non-relativistic atoms, the low-energy excitations of superfluid, i.e. the phonons, are said to be "relativistic". I know that the superfluid phonons do have the linear dispersion $\omega = c k$, and the low-energy effective theory is Lorentz invariant. Also the superfluid is inviscid, so that the aether has no drag force. But I still doubt that the special relativity can really emerge in the superfluid. Because I can not show the following statement: The velocity of sound in the superfluid is the same for all inertial observers, regardless of their relative motion to the superfluid. I believe this should be a fundamental feature of relativity, but I do not see how to prove it. The velocity addition formula for sound is not the same as that for light. So I am very confused when people say that the phonons are relativistic. Any understanding of the emergent relativity in superfluid would be appreciated.

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The superfluid phonons have the linear dispersion $\omega=ck$, and the low-energy effective theory is Lorentz invariant. So the superfluid is an example of emergent Lorentz symmetry. The statement: The velocity of sound in the superfluid is the same for all inertial observers, regardless of their relative motion to the superfluid is valid if the clock and ruler are make by low energy phonons.
But in "real world" the constant speed of light does not require that the clock and ruler to be made by photon, right? Also, what about a system that has two kind of phonons, with speed $c_1$ and $c_2$ respectively? I tend to believe the phonons do have linear dispersions, but it does not reflect a relativistic nature of the emergent spcetime. – pathintegral Jul 12 at 1:24