The Null Energy Condition through the Lens of Optimal Transport: Smooth and Non-Smooth Spacetimes
Abstract
We survey recent developments on the null energy condition (NEC) from the perspective of optimal transport, with emphasis on non-smooth Lorentzian geometry. After recalling the role of classical energy conditions in general relativity, we discuss the relation between Ricci curvature and entropy convexity in optimal transport. The main focus is the authors' optimal-transport characterization of the NEC. In the smooth setting, the NEC is equivalent to displacement concavity of the Shannon entropy power along suitable null-geodesic transport plans supported on null hypersurfaces and measured with respect to rigged measures. We review the geometric ingredients of this characterization and applications, including a weighted Hawking area theorem. We then turn to the synthetic theory introduced in a second paper of the authors. Synthetic null hypersurfaces are defined using achronal boundaries, gauge functions, and reference measures in topological causal spaces. In this framework, the displacement-concavity characterization becomes a synthetic curvature condition, NCe(N), meaningful without differentiable structure. We discuss compatibility with the smooth theory, invariance under natural changes of gauge and reference measure, and stability under convergence. Finally, we present a synthetic version of Penrose's singularity theorem for continuous Lorentzian metrics. These results indicate that geometric and causal consequences of the NEC persist beyond the smooth setting and highlight optimal transport as a framework connecting curvature, causality, and singularity formation in Lorentzian geometry.
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BibTeX
@misc{cavalletti2026null,
title = {{The Null Energy Condition through the Lens of Optimal Transport: Smooth and Non-Smooth Spacetimes}},
author = {Fabio Cavalletti and Davide Manini and Andrea Mondino},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16942},
eprint = {2609.16942},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16942},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Null Energy Condition through the Lens of Optimal Transport: Smooth and Non-Smooth Spacetimes
AU - Fabio Cavalletti
AU - Davide Manini
AU - Andrea Mondino
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16942
UR - https://arxiv.org/abs/2609.16942
AB - We survey recent developments on the null energy condition (NEC) from the perspective of optimal transport, with emphasis on non-smooth Lorentzian geometry. After recalling the role of classical energy conditions in general relativity, we discuss the relation between Ricci curvature and entropy convexity in optimal transport. The main focus is the authors' optimal-transport characterization of the NEC. In the smooth setting, the NEC is equivalent to displacement concavity of the Shannon entropy power along suitable null-geodesic transport plans supported on null hypersurfaces and measured with respect to rigged measures. We review the geometric ingredients of this characterization and applications, including a weighted Hawking area theorem. We then turn to the synthetic theory introduced in a second paper of the authors. Synthetic null hypersurfaces are defined using achronal boundaries, gauge functions, and reference measures in topological causal spaces. In this framework, the displacement-concavity characterization becomes a synthetic curvature condition, NC^e(N), meaningful without differentiable structure. We discuss compatibility with the smooth theory, invariance under natural changes of gauge and reference measure, and stability under convergence. Finally, we present a synthetic version of Penrose's singularity theorem for continuous Lorentzian metrics. These results indicate that geometric and causal consequences of the NEC persist beyond the smooth setting and highlight optimal transport as a framework connecting curvature, causality, and singularity formation in Lorentzian geometry.
ER -