On geodesics in space-times with a foliation structure: A spectral geometry approach

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.