Concavity of spacetimes

TL;DR

The paper investigates the local concavity of time separation in Berwald spacetimes as a Lorentzian analogue of Busemann convexity, showing that nonnegative flag curvature in timelike directions ($\mathbf{K} \ge 0$) is equivalent to local concavity, local timelike concavity, and the convexity of future/past capsules. It introduces convex capsules as a structural proxy for curvature bounds and proves their equivalence to curvature conditions, providing new characterizations that extend to Lorentzian manifolds. The work advances synthetic Lorentzian geometry by linking curvature to causal-geometric convexity and offering a framework that could underpin future developments in Lorentzian optimal transport and curvature-dimension theory, while also outlining open problems in non-Berwald settings and in defining Lorentzian barycenters. Overall, these results contribute new, technically precise connections between curvature, causality, and convexity in Lorentz--Finsler geometry with potential broad-ranging implications for synthetic spacetime analysis.

Abstract

Motivated by recent breathtaking progress in the synthetic study of Lorentzian geometry, we investigate the local concavity of time separation functions on Finsler spacetimes as a Lorentzian counterpart to Busemann's convexity in metric geometry. We show that a Berwald spacetime is locally concave if and only if its flag curvature is nonnegative in timelike directions. We also give another characterization of nonnegative flag curvature by the convexity of future (or past) capsules, inspired by Kristály--Kozma's result in the positive definite case. These characterizations are new even for Lorentzian manifolds.

Figures (2)

• Figure 1: The convex past capsule associated to a geodesic $\gamma$.
• Figure 2: The geodesic variation $\sigma$ and a tangent hyperbola.

Theorems & Definitions (37)

• Theorem 1.1: Characterizations of concavity
• Corollary 1.2: Lorentzian case
• Definition 2.1: Lorentz--Finsler structures
• Definition 2.2: Finsler spacetimes
• Definition 2.3: Berwald spacetimes
• Remark 2.4: Metrizability
• Remark 2.5: Sign of $\mathbf{K}$
• Definition 2.6: Causality conditions
• Definition 3.1: Concavity
• Remark 3.2: Causal character of geodesics