Optimal transport on null hypersurfaces and the null energy condition
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915592674476032 |
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| author | Cavalletti, Fabio Manini, Davide Mondino, Andrea |
| author_facet | Cavalletti, Fabio Manini, Davide Mondino, Andrea |
| contents | 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_08986 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimal transport on null hypersurfaces and the null energy condition Cavalletti, Fabio Manini, Davide Mondino, Andrea Differential Geometry Mathematical Physics 83C05 (primary) 49Q22 (secondary) 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). |
| title | Optimal transport on null hypersurfaces and the null energy condition |
| topic | Differential Geometry Mathematical Physics 83C05 (primary) 49Q22 (secondary) |
| url | https://arxiv.org/abs/2408.08986 |