Exploring the Causal Structures of Almost Commutative Geometries

Nicolas Franco, Michał Eckstein

Published 2013-10-30, updated 2014-01-28Version 2

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.