Lorentzian approach to noncommutative geometry

TL;DR

This thesis explores extending Alain Connes' noncommutative geometry to Lorentzian manifolds, introducing a new distance function and axioms for Lorentzian spectral triples to incorporate time in noncommutative frameworks.

Contribution

It proposes the first steps towards a Lorentzian generalization of noncommutative geometry, including a new distance function and axioms for temporal Lorentzian spectral triples.

Findings

A Lorentzian distance function using a global timelike eikonal condition

Initial axioms for a temporal Lorentzian spectral triple

Framework for incorporating global time in noncommutative geometry

Abstract

This thesis concerns the research on a Lorentzian generalization of Alain Connes' noncommutative geometry. In the first chapter, we present an introduction to noncommutative geometry within the context of unification theories. The second chapter is dedicated to the basic elements of noncommutative geometry as the noncommutative integral, the Riemannian distance function and spectral triples. In the last chapter, we investigate the problem of the generalization to Lorentzian manifolds. We present a first step of generalization of the distance function with the use of a global timelike eikonal condition. Then we set the first axioms of a temporal Lorentzian spectral triple as a generalization of a pseudo-Riemannian spectral triple together with a notion of global time in noncommutative geometry.