  Astrophysical tests of Planck-suppressed Lorentz violations had been
extensively studied in recent years and very stringent constraints have been
obtained within the framework of effective field theory. There are however
still some unresolved theoretical issues, in particular regarding the so called
"naturalness problem" - which arises when postulating that Planck-suppressed
Lorentz violations arise only from operators with mass dimension greater than
four in the Lagrangian. In the work presented here we shall try to address this
problem by looking at a condensed-matter analogue of the Lorentz violations
considered in quantum gravity phenomenology. Specifically, we investigate the
class of two-component BECs subject to laser-induced transitions between the
two components, and we show that this model is an example for Lorentz
invariance violation due to ultraviolet physics. We shall show that such a
model can be considered to be an explicit example high-energy Lorentz
violations where the ``naturalness problem'' does not arise.
