Space of Timelike Directions and Curvature Bounds

TL;DR

This paper extends synthetic curvature bounds to Lorentzian length spaces, showing that under timelike sectional curvature bounds, the space of directions exists with bounded curvature, and tangent cones form Lorentzian length spaces with non-positive curvature.

Contribution

It introduces $ ext{ extepsilon}$-$ ext{ extmu}$ timelike sectional curvature bounds and establishes the existence and properties of the space of directions and tangent cones in Lorentzian length spaces.

Findings

Space of directions exists with curvature bounded above by -1.

Tangent cones form Lorentzian length spaces with curvature bounded above by 0.

Provides a synthetic framework for curvature and geodesics in Lorentzian geometry.

Abstract

We investigate the consequences of timelike sectional curvature bounds in Lorentzian length spaces for the existence and structure of the space of directions at a point. It is established that, under upper timelike sectional curvature bounds, the space of directions exists and is itself a metric space with curvature bounded above by $-1$. Furthermore, the metric cone over the space of directions, which canonically models the tangent space at a given point, is shown to constitute a Lorentzian length space with timelike sectional curvature bounded above by $0$. To do this, we introduce the notion of $\epsilon$-$\mu$ timelike sectional curvature bounds, which are compatible with pre-existing synthetic curvature conditions. These results extend the comparison-geometric framework to the Lorentzian setting, providing a synthetic characterization of geodesics, tangent cones, and curvature under causal constraints.