Timelike Ollivier-Ricci curvature
Abstract
We introduce a codimension one construction of coarse Ricci curvature in Lorentzian geometry. Using the $1$-Lorentz-Wasserstein distance ℓ1, we compare probability measures supported on small spacelike hypersurfaces through nearby events and recover, in a precise asymptotic regime, the ambient Ricci curvature in future-directed unit timelike directions, with a universal dimensional prefactor reflecting this codimension one construction. The construction echoes the Raychaudhuri equation, which relates the evolution of spatial volume expansion along timelike geodesics to Ricci curvature. Our quantitative estimates rely on transport maps with controlled displacement, constructed through a Moser-type flow. At leading order, the slice construction is insensitive to smooth weights. By smearing the spacelike slices into thin timelike tubes and calibrating the temporal and spatial scales, we recover the timelike Bakry-Émery tensor Ric+Hess V associated with the weighted reference measure smashm=e−V volg.
Related
- Idleness Functions for Ollivier-Ricci Curvature on Hypergraphs
- Ollivier-Ricci curvature for causal sets
- Ollivier--Ricci Idleness Functions and Edge-Connectivity of Hypergraphs
- Ollivier curvature, isoperimetry, concentration, and log-Sobolev inequality
- Almost-splitting of spacetimes under timelike sectional curvature bounds
BibTeX
@misc{braun2026timelike,
title = {{Timelike Ollivier-Ricci curvature}},
author = {Mathias Braun and Xue-Mei Li},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18664},
eprint = {2609.18664},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18664},
}
RIS
TY - UNPB
TI - Timelike Ollivier-Ricci curvature
AU - Mathias Braun
AU - Xue-Mei Li
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18664
UR - https://arxiv.org/abs/2609.18664
AB - We introduce a codimension one construction of coarse Ricci curvature in Lorentzian geometry. Using the $1$-Lorentz-Wasserstein distance ℓ_1, we compare probability measures supported on small spacelike hypersurfaces through nearby events and recover, in a precise asymptotic regime, the ambient Ricci curvature in future-directed unit timelike directions, with a universal dimensional prefactor reflecting this codimension one construction. The construction echoes the Raychaudhuri equation, which relates the evolution of spatial volume expansion along timelike geodesics to Ricci curvature. Our quantitative estimates rely on transport maps with controlled displacement, constructed through a Moser-type flow. At leading order, the slice construction is insensitive to smooth weights. By smearing the spacelike slices into thin timelike tubes and calibrating the temporal and spatial scales, we recover the timelike Bakry-Émery tensor Ric+Hess V associated with the weighted reference measure smashm=e^−V vol_g.
ER -