A nonlinear d'Alembert comparison theorem and causal differential calculus on metric measure spacetimes
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arXiv
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| Autori principali: | , , , , , , , |
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| Natura: | Preprint |
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2024
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| _version_ | 1866909544177729536 |
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| author | Beran, Tobias Braun, Mathias Calisti, Matteo Gigli, Nicola McCann, Robert J. Ohanyan, Argam Rott, Felix Sämann, Clemens |
| author_facet | Beran, Tobias Braun, Mathias Calisti, Matteo Gigli, Nicola McCann, Robert J. Ohanyan, Argam Rott, Felix Sämann, Clemens |
| contents | We introduce a variational first-order Sobolev calculus on metric measure spacetimes. The key object is the maximal weak subslope of an arbitrary causal function, which plays the role of the (Lorentzian) modulus of its differential. It is shown to satisfy certain chain and Leibniz rules, certify a locality property, and be compatible with its smooth analog. In this setup, we propose a quadraticity condition termed infinitesimal Minkowskianity, which singles out genuinely Lorentzian structures among Lorentz-Finsler spacetimes. Moreover, we establish a comparison theorem for a nonlinear yet elliptic $p$-d'Alembertian in a weak form under the timelike measure contraction property. As a particular case, this extends Eschenburg's classical estimate past the timelike cut locus. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_15968 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A nonlinear d'Alembert comparison theorem and causal differential calculus on metric measure spacetimes Beran, Tobias Braun, Mathias Calisti, Matteo Gigli, Nicola McCann, Robert J. Ohanyan, Argam Rott, Felix Sämann, Clemens Differential Geometry General Relativity and Quantum Cosmology Mathematical Physics Metric Geometry 51K10, 83C75 (Primary), 35J92, 35Q75, 49Q22, 53C21, 53C50 (Secondary) We introduce a variational first-order Sobolev calculus on metric measure spacetimes. The key object is the maximal weak subslope of an arbitrary causal function, which plays the role of the (Lorentzian) modulus of its differential. It is shown to satisfy certain chain and Leibniz rules, certify a locality property, and be compatible with its smooth analog. In this setup, we propose a quadraticity condition termed infinitesimal Minkowskianity, which singles out genuinely Lorentzian structures among Lorentz-Finsler spacetimes. Moreover, we establish a comparison theorem for a nonlinear yet elliptic $p$-d'Alembertian in a weak form under the timelike measure contraction property. As a particular case, this extends Eschenburg's classical estimate past the timelike cut locus. |
| title | A nonlinear d'Alembert comparison theorem and causal differential calculus on metric measure spacetimes |
| topic | Differential Geometry General Relativity and Quantum Cosmology Mathematical Physics Metric Geometry 51K10, 83C75 (Primary), 35J92, 35Q75, 49Q22, 53C21, 53C50 (Secondary) |
| url | https://arxiv.org/abs/2408.15968 |