Concavity of spacetimes

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.

20 pages, 2 figures. Final version

References in corpus (9)

• Convex neighborhoods for Lipschitz connections and sprays
• An optimal transport formulation of the Einstein equations of general relativity
• Optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds and applications
• Raychaudhuri equation and singularity theorems in Finsler spacetimes
• The splitting theorem for globally hyperbolic Lorentzian length spaces with non-negative timelike curvature
• On the non metrizability of Berwald Finsler spacetimes
• Hyperbolic angles in Lorentzian length spaces and timelike curvature bounds
• Geometry of weighted Lorentz-Finsler manifolds I: Singularity theorems
• Geometry of weighted Lorentz-Finsler manifolds II: A splitting theorem