Causal structure for noncommutative geometry

Noncommutative Geometry à la Connes offers a promising framework for models of fundamental interactions. To guarantee the correct signature, the theory of Lorentzian spectral triples has been developed. We will briefly summarise its main elements and show that it can accommodate a sensible notion of causality understood as a partial order relation on the space of states on an algebra. For almost-commutative algebras of the form $C^\infty \otimes \A_F$, with $\A_F$ being finite-dimensional, the space of (pure) states is a simple product of space-time $\M$ and an internal space. The exploration of causal structures in this context leads to a surprising conclusion: The motion in both space-time and internal space is restricted by a "finite speed of light" constraint. We will present this phenomena on 2 simple toy-models.