On the definition of spacetimes in Noncommutative Geometry, Part I

TL;DR

This paper extends Connes' spectral triple framework to Lorentzian manifolds, characterizing spacetime signatures via noncommutative geometry tools, laying groundwork for a broader noncommutative spacetime theory.

Contribution

It introduces the concept of spectral spacetime in the commutative case, providing a foundation for Lorentzian noncommutative geometry in subsequent work.

Findings

Characterization of metric signature via a time-orientation 1-form

Krein product on spinor fields derived from geometric data

Focus on space and time oriented even-dimensional spin manifolds

Abstract

In this two-part paper we propose an extension of Connes' notion of even spectral triple to the Lorentzian setting. This extension, which we call a spectral spacetime, is discussed in part II where several natural examples are given which are not covered by the previous approaches to the problem. Part I only deals with the commutative and continuous case of a manifold. It contains all the necessary material for the generalization to come in part II, namely the characterization of the signature of the metric 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. Thus, only space and time oriented spin manifolds of even dimension are considered for a noncommutative generalization in this approach. We observe that these are precisely the kind of manifolds on which the modern theories of spacetime and matter are defined.