Generalized cones as Lorentzian length spaces: Causality, curvature, and singularity theorems
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866912007090864128 |
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| author | Alexander, Stephanie B. Graf, Melanie Kunzinger, Michael Sämann, Clemens |
| author_facet | Alexander, Stephanie B. Graf, Melanie Kunzinger, Michael Sämann, Clemens |
| contents | We study generalizations of Lorentzian warped products with one-dimensional base of the form $I\times_f X$, where $I$ is an interval, $X$ is a length space and $f$ is a positive continuous function. These generalized cones furnish an important class of Lorentzian length spaces in the sense of [Kunzinger, Sämann; Ann. Glob. Anal. Geom. 54(3):399--447, 2018], displaying optimal causality properties that allow for explicit descriptions of all underlying notions. In addition, synthetic sectional curvature bounds of generalized cones are directly related to metric curvature bounds of the fiber $X$. The interest in such spaces comes both from metric geometry and from General Relativity, where warped products underlie important cosmological models (FLRW spacetimes). Moreover, we prove singularity theorems for these spaces, showing that non-positive lower timelike curvature bounds imply the existence of incomplete timelike geodesics. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1909_09575 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Generalized cones as Lorentzian length spaces: Causality, curvature, and singularity theorems Alexander, Stephanie B. Graf, Melanie Kunzinger, Michael Sämann, Clemens Metric Geometry General Relativity and Quantum Cosmology Mathematical Physics Differential Geometry 51K10, 53C23, 53C50, 53B30, 53C80 We study generalizations of Lorentzian warped products with one-dimensional base of the form $I\times_f X$, where $I$ is an interval, $X$ is a length space and $f$ is a positive continuous function. These generalized cones furnish an important class of Lorentzian length spaces in the sense of [Kunzinger, Sämann; Ann. Glob. Anal. Geom. 54(3):399--447, 2018], displaying optimal causality properties that allow for explicit descriptions of all underlying notions. In addition, synthetic sectional curvature bounds of generalized cones are directly related to metric curvature bounds of the fiber $X$. The interest in such spaces comes both from metric geometry and from General Relativity, where warped products underlie important cosmological models (FLRW spacetimes). Moreover, we prove singularity theorems for these spaces, showing that non-positive lower timelike curvature bounds imply the existence of incomplete timelike geodesics. |
| title | Generalized cones as Lorentzian length spaces: Causality, curvature, and singularity theorems |
| topic | Metric Geometry General Relativity and Quantum Cosmology Mathematical Physics Differential Geometry 51K10, 53C23, 53C50, 53B30, 53C80 |
| url | https://arxiv.org/abs/1909.09575 |