  Several approaches to quantum gravity suggest that Lorentz invariance will be
broken at high energy. This can lead to modified dispersion relations for wave
propagation, which can be concretely realized in effective field theories where
the equation of motion involves higher order spatial derivatives in a preferred
frame. We consider such a model in the presence of a finite cavity whose walls
follow parallel inertial trajectories of speed $v$ with respect to the
preferred frame. We find evidence that when the cavity wall speed exceeds the
phase velocity, the system becomes classically unstable. For dispersion
relations that do not lead to an instability, the energy levels of the cavity
are non-trivial functions of $v$. In other words, an observer could in
principle measure their velocity with respect to the preferred frame by
studying the energy spectra of a quantum cavity, which is a stark violation of
the principle of relativity. We also find that the energy levels of the cavity
become infinitely large as its velocity approaches light speed.
