Aspects of noncommutative Lorentzian geometry for globally hyperbolic spacetimes

TL;DR

This paper extends Connes' noncommutative geometry framework to Lorentzian spacetimes, introducing a noncommutative causal structure and Lorentzian distance, with proofs of smoothness and a new algebraic approach.

Contribution

It develops a Lorentzian version of noncommutative geometry using $C^*$-algebras, causal cones, and loci, generalizing spacetime structure to a noncommutative setting.

Findings

Generalized Lorentzian distance as an operator in noncommutative setting

Defined noncommutative causal order among loci

Extended the concept of events to probability measures

Abstract

Connes' functional formula of the Riemannian distance is generalized to the Lorentzian case using the so-called Lorentzian distance, the d'Alembert operator and the causal functions of a globally hyperbolic spacetime. As a step of the presented machinery, a proof of the almost-everywhere smoothness of the Lorentzian distance considered as a function of one of the two arguments is given. Afterwards, using a $C^*$-algebra approach, the spacetime causal structure and the Lorentzian distance are generalized into noncommutative structures giving rise to a Lorentzian version of part of Connes' noncommutative geometry. The generalized noncommutative spacetime consists of a direct set of Hilbert spaces and a related class of $C^*$-algebras of operators. In each algebra a convex cone made of self-adjoint elements is selected which generalizes the class of causal functions. The generalized events, called {\em loci}, are realized as the elements of the inductive limit of the spaces of the algebraic states on the $C^*$-algebras. A partial-ordering relation between pairs of loci generalizes the causal order relation in spacetime. A generalized Lorentz distance of loci is defined by means of a class of densely-defined operators which play the r\^ole of a Lorentzian metric. Specializing back the formalism to the usual globally hyperbolic spacetime, it is found that compactly-supported probability measures give rise to a non-pointwise extension of the concept of events.