  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 temporal
Lorentzian spectral triple over a Moyal--Minkowski spacetime. We show that,
when time is commutative, the algebra can be extended to unbounded elements.
Using such an extension, it is possible to define a Lorentzian distance formula
between pure states with a well-defined noncommutative formulation.
