Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 6 (2010), 064, 11 pages      arXiv:1003.5651      https://doi.org/10.3842/SIGMA.2010.064
Contribution to the Special Issue “Noncommutative Spaces and Fields”

Global Eikonal Condition for Lorentzian Distance Function in Noncommutative Geometry

Nicolas Franco
GAMASCO, Department of Mathematics, University of Namur FUNDP, 8 Rempart de la Vierge, B-5000 Namur, Belgium

Received March 30, 2010, in final form August 06, 2010; Published online August 17, 2010

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.

Key words: noncommutative geometry; Lorentzian distance; eikonal inequality.

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