Title: On the definition of spacetimes in noncommutative geometry: Part I
Abstract: In this paper we consider semi-Riemannian time and space oriented manifolds of even dimension, and characterize the Lorentzian and antilorentzian signatures in terms of a time-orientation 1-form and a natural Krein product on spinor fields. It turns out that all the data available in Noncommutative Geometry (the algebra of functions, the Krein space of spinor fields, the representation of the algebra on it, the Dirac operator, charge conjugation and chirality), but nothing more, play a role in this characterization. It thus yields a possible definition extending Connes’ notion of even spectral triple to the Lorentzian setting.