  Motivated by the Horava-Lifshitz type theories, we study the physical motion
of matter coupled to a foliated geometry in non-diffeomorphism invariant way.
We use the concept of a spectral action as a guiding principle in writing down
the matter action. Based on the deformed Dirac operator compatible with the
reduced symmetry - foliation preserving diffeomorphisms, this approach provides
a natural generalization of the minimal coupling. Focusing on the IR version of
the Dirac operator, we derive the physical motion of a test particle and
discuss in what sense it still can be considered as a geodesic motion for some
modified geometry. We show that the apparatus of non-commutative geometry could
be very efficient in the study of matter coupled to the Horava-Lifshitz
gravity.
