September 02, 2014
We present the notion of temporal Lorentzian spectral triple which is an extension of the notion of pseudo-Riemannian spectral triple with a way to ensure that the signature of the metric is Lorentzian. A temporal Lorentzian spectral triple corresponds to a specific 3+1 decomposition of a possibly noncommutative Lorentzian space. This structure introduces a notion of global time in noncommutative geometry. As an example, we construct a noncommutative Minkowsky spacetime with the use of a degenerate Moyal product.
Keywords:noncommutative geometry, Lorentzian geometry, spectral triples