# Exploring the Causal Structures of Almost Commutative Geometries

January 28, 2014

We investigate the causal relations in the space of states of almost
commutative Lorentzian geometries. We fully describe the causal structure of a
simple model based on the algebra $\mathcal{S}(\mathbb{R}^{1,1}) \otimes
M_2(\mathbb{C})$, which has a non-trivial space of internal degrees of freedom.
It turns out that the causality condition imposes restrictions on the motion in
the internal space. Moreover, we show that the requirement of causality favours
a unitary evolution in the internal space.

Keywords:

noncommutative geometry, causality, lorentzian spectral triples