Non Hilbertian (Lorentzian) Length Spaces

TL;DR

This paper explores Lorentzian length spaces inspired by $L^p$ spaces, demonstrating that non-Hilbertian norms lack sectional curvature bounds and analyzing their implications for Lorentzian Gromov-Hausdorff convergence.

Contribution

It introduces Lorentzian $L^p$ spaces as examples of non-Hilbertian length spaces and investigates their curvature properties and convergence behavior.

Findings

Normed spaces without inner products lack sectional curvature bounds.

Lorentzian $L^p$ spaces serve as examples in Gromov-Hausdorff convergence analysis.

Unbounded curvature and geodesic irregularities are not preserved in limits.

Abstract

In this note, the idea of finite dimensional $L^p$ spaces is transferred to Lorentzian length spaces to provide an example that is locally nowhere Minkowskian. Looking at the sectional curvature bounds of this example leads to the more general statement that normed spaces in which the norm does not come from an inner product, have no sectional curvature bounds. This statement holds in the Riemannian and Lorentzian cases. In addition, the Lorentzian $L^p$ space can be used as an example in the context of Lorentzian Gromov-Hausdorff convergence, to show that unbounded sectional curvature or geodesic regularity is in general not preserved in the GH limit, and as an example of a sequence of uniform bounded length spaces which are not GH pre-compact.