Gromov-Hausdorff metrics and dimensions of Lorentzian length spaces
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866912265905635328 |
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| author | Müller, Olaf |
| author_facet | Müller, Olaf |
| contents | We construct analoga of Gromov-Hausdorff space for Lorentzian distances and show a Gromov precompactness result for one of them. After calculating the Dushnik-Miller dimension of Minkowski spaces (of manifold dimension larger than 2) to be countable infinity, we define a dimension for ordered sets recovering the correct manifold dimension, obtain an obstruction for existence of injective monotonous maps between Lorentzian length spaces, induce functorial pseudo-metrics on Cauchy subsets that in the spacetime case coincide with the Riemannian ones, and prove existence of anti-Lipschitz Cauchy functions with a given Cauchy zero locus, a fundamental ingredient for the Sormani-Vega null distance. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_12736 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Gromov-Hausdorff metrics and dimensions of Lorentzian length spaces Müller, Olaf Differential Geometry 53C23, 53C80, 51F99 We construct analoga of Gromov-Hausdorff space for Lorentzian distances and show a Gromov precompactness result for one of them. After calculating the Dushnik-Miller dimension of Minkowski spaces (of manifold dimension larger than 2) to be countable infinity, we define a dimension for ordered sets recovering the correct manifold dimension, obtain an obstruction for existence of injective monotonous maps between Lorentzian length spaces, induce functorial pseudo-metrics on Cauchy subsets that in the spacetime case coincide with the Riemannian ones, and prove existence of anti-Lipschitz Cauchy functions with a given Cauchy zero locus, a fundamental ingredient for the Sormani-Vega null distance. |
| title | Gromov-Hausdorff metrics and dimensions of Lorentzian length spaces |
| topic | Differential Geometry 53C23, 53C80, 51F99 |
| url | https://arxiv.org/abs/2209.12736 |