Optimal transport and timelike lower Ricci curvature bounds on Finsler spacetimes
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917695573721088 |
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| author | Braun, Mathias Ohta, Shin-ichi |
| author_facet | Braun, Mathias Ohta, Shin-ichi |
| contents | We prove that a Finsler spacetime endowed with a smooth reference measure whose induced weighted Ricci curvature $\smash{\mathrm{Ric}_N}$ is bounded from below by a real number $K$ in every timelike direction satisfies the timelike curvature-dimension condition $\smash{\mathrm{TCD}_q(K,N)}$ for all $q\in (0,1)$. A nonpositive-dimensional version ($N \le 0$) of this result is also shown. Our discussion is based on the solvability of the Monge problem with respect to the $q$-Lorentz-Wasserstein distance as well as the characterization of $q$-geodesics of probability measures. One consequence of our work is the sharp timelike Brunn-Minkowski inequality in the Lorentz-Finsler case. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_04389 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Optimal transport and timelike lower Ricci curvature bounds on Finsler spacetimes Braun, Mathias Ohta, Shin-ichi Differential Geometry General Relativity and Quantum Cosmology Metric Geometry 53C50, 53C60, 58E10, 83C99 We prove that a Finsler spacetime endowed with a smooth reference measure whose induced weighted Ricci curvature $\smash{\mathrm{Ric}_N}$ is bounded from below by a real number $K$ in every timelike direction satisfies the timelike curvature-dimension condition $\smash{\mathrm{TCD}_q(K,N)}$ for all $q\in (0,1)$. A nonpositive-dimensional version ($N \le 0$) of this result is also shown. Our discussion is based on the solvability of the Monge problem with respect to the $q$-Lorentz-Wasserstein distance as well as the characterization of $q$-geodesics of probability measures. One consequence of our work is the sharp timelike Brunn-Minkowski inequality in the Lorentz-Finsler case. |
| title | Optimal transport and timelike lower Ricci curvature bounds on Finsler spacetimes |
| topic | Differential Geometry General Relativity and Quantum Cosmology Metric Geometry 53C50, 53C60, 58E10, 83C99 |
| url | https://arxiv.org/abs/2305.04389 |