Global Eikonal Condition for Lorentzian Distance Function in Noncommutative Geometry

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.

Special Issue on Noncommutative Spaces and Field

References in corpus (4)

• On smooth Cauchy hypersurfaces and Geroch's splitting theorem
• Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes
• On Noncommutative and semi-Riemannian Geometry
• Equivariant Lorentzian Spectral Triples

Cited by in corpus (7)

• Temporal Lorentzian Spectral Triples
• Noncommutative geometry, Lorentzian structures and causality
• Length and distance on a quantum space
• Noncommutative geometrical origin of the energy-momentum dispersion relation
• Spectral Distance on Lorentzian Moyal Plane
• Families of spectral triples and foliations of space(time)
• The global properties of the finiteness and continuity of the Lorentzian distance