Global Eikonal Condition for Lorentzian Distance Function in Noncommutative Geometry

TL;DR

This paper extends Connes' noncommutative Riemannian distance framework to Lorentzian geometry, demonstrating that a global timelike eikonal condition suffices for a path-independent Lorentzian distance in globally hyperbolic spacetimes.

Contribution

It introduces a global eikonal condition for Lorentzian distance, generalizing the Riemannian case within noncommutative geometry.

Findings

A single global timelike eikonal condition suffices in globally hyperbolic spacetimes.

The approach constructs a path-independent Lorentzian distance function.

Generalization of noncommutative geometric methods to Lorentzian settings.

Abstract

Connes' noncommutative Riemannian distance formula is constructed in two steps, the first one being the construction of a path-independent geometrical functional using a global constraint on continuous functions. This paper generalizes this first step to Lorentzian geometry. We show that, in a globally hyperbolic spacetime, a single global timelike eikonal condition is sufficient to construct a path-independent Lorentzian distance function.