# An algebraic formulation of causality for noncommutative geometry

December 20, 2012

We propose an algebraic formulation of the notion of causality for spectral
triples corresponding to globally hyperbolic manifolds with a well defined
noncommutative generalization. The causality is given by a specific cone of
Hermitian elements respecting an algebraic condition based on the Dirac
operator and a fundamental symmetry. We prove that in the commutative case the
usual notion of causality is recovered. We show that, when the dimension of the
manifold is even, the result can be extended in order to have an algebraic
constraint suitable for a Lorentzian distance formula.

Keywords:

noncommutative geometry, causality, lorentzian spectral triples