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Optimal transport on null hypersurfaces and the null energy condition

Fabio Cavalletti, Davide Manini, Andrea Mondino

TL;DR

The paper develops a diffused optimal transport framework for null hypersurfaces in Lorentzian manifolds, overcoming degenerate costs by employing a rigged volume to define entropy-based energy notions. It proves that null energy conditions can be characterized via displacement convexity of the Boltzmann–Shannon entropy along monotone null-geodesic transports, bridging synthetic Ricci-type bounds with classical NEC in smooth settings. The work establishes existence and uniqueness of monotone transport plans on null hypersurfaces, shows NC^1(N) ⇔ NC^e(N) ⇔ NEC under appropriate regularity, and proves stability under geometric limits. Applications include weighted versions of the light-cone and Hawking area theorems, with rigidity statements that connect equality cases to isometries with Minkowski space and to static horizon structures, paving the way for extensions to non-smooth Lorentzian spaces.

Abstract

The goal of the present work is to study optimal transport on null hypersurfaces inside Lorentzian manifolds. The challenge here is that optimal transport along a null hypersurface is completely degenerate, as the cost takes only the two values $0$ and $+\infty$. The tools developed in the manuscript enable to give an optimal transport characterization of the null energy condition (namely, non-negative Ricci curvature in the null directions) for Lorentzian manifolds in terms of convexity properties of the Boltzmann--Shannon entropy along null-geodesics of probability measures. We obtain as applications: a stability result under convergence of spacetimes, a comparison result for null-cones, and the Hawking area theorem (both in sharp form, for possibly weighted measures, and with apparently new rigidity statements).

Optimal transport on null hypersurfaces and the null energy condition

TL;DR

The paper develops a diffused optimal transport framework for null hypersurfaces in Lorentzian manifolds, overcoming degenerate costs by employing a rigged volume to define entropy-based energy notions. It proves that null energy conditions can be characterized via displacement convexity of the Boltzmann–Shannon entropy along monotone null-geodesic transports, bridging synthetic Ricci-type bounds with classical NEC in smooth settings. The work establishes existence and uniqueness of monotone transport plans on null hypersurfaces, shows NC^1(N) ⇔ NC^e(N) ⇔ NEC under appropriate regularity, and proves stability under geometric limits. Applications include weighted versions of the light-cone and Hawking area theorems, with rigidity statements that connect equality cases to isometries with Minkowski space and to static horizon structures, paving the way for extensions to non-smooth Lorentzian spaces.

Abstract

The goal of the present work is to study optimal transport on null hypersurfaces inside Lorentzian manifolds. The challenge here is that optimal transport along a null hypersurface is completely degenerate, as the cost takes only the two values and . The tools developed in the manuscript enable to give an optimal transport characterization of the null energy condition (namely, non-negative Ricci curvature in the null directions) for Lorentzian manifolds in terms of convexity properties of the Boltzmann--Shannon entropy along null-geodesics of probability measures. We obtain as applications: a stability result under convergence of spacetimes, a comparison result for null-cones, and the Hawking area theorem (both in sharp form, for possibly weighted measures, and with apparently new rigidity statements).
Paper Structure (24 sections, 48 theorems, 238 equations)

This paper contains 24 sections, 48 theorems, 238 equations.

Table of Contents

  1. Introduction
  2. Conventions and preliminaries
  3. General facts of Lorentzian geometry
  4. Null hypersurfaces
  5. The rigging technique
  6. Local Parameterization
  7. Volume measures on Null hypersurfaces
  8. Jacobi fields in null hypersurfaces
  9. Local expression of the rigged Volume
  10. Independence from local cross-sections
  11. Volume distortion of the (weighted) rigged measure
  12. Weighted rigged measure
  13. Rigged measure and Ricci curvature: the NC1(N) condition
  14. Optimal Transport inside null hypersurfaces
  15. Existence and uniqueness of monotone plans along $H$.
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $(M^{n},g)$ be a Lorentzian manifold, let $H$ be a causal null hypersurface admitting a space-like and acausal cross-section $S$, and a null-geodesic vector field $L$. Then the following representation formula holds where:

Theorems & Definitions (127)

  • Theorem 1.1
  • Definition 1.2: $\mathsf{NC}^{1}(n)$ condition
  • Definition 1.3: Transport inside a null hypersurface
  • Definition 1.4: Null-geodesic dynamical transport plan
  • Definition 1.5: Monotone couplings and plans
  • Theorem 1.6
  • Definition 1.7: The $\mathsf{NC}^e(n)$ condition
  • Proposition 2.1: Pag. 45 in HawEll
  • Remark 2.2
  • Proposition 2.3
  • ...and 117 more