Pseudo-timelike loops in signature changing semi-Riemannian manifolds with a transverse radical
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866915888188358656 |
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| author | Rieger, N. E. Hasse, W. |
| author_facet | Rieger, N. E. Hasse, W. |
| contents | In 1983, Hartle and Hawking proposed the no-boundary proposal, suggesting that the universe has no beginning in the sense of a spacetime singularity or boundary. Nevertheless, there is an origin of time. Mathematically, this involves signature-type changing manifolds in which a Riemannian region smoothly transitions to a Lorentzian region across the hypersurface $\mathcal{H}$ where time begins.
We develop a coherent framework for signature changing manifolds with a degenerate yet smooth metric. Established Lorentzian tools and results are then adapted to this setting, and new definitions are introduced that carry unforeseen causal implications. A noteworthy consequence is the presence of locally time-reversing loops through every point on the hypersurface. Imposing global hyperbolicity on the Lorentzian region, we prove that for every point $p \in M$ there exists a pseudo-timelike loop self-intersecting at $p$. Equivalently, $M$ always admits a closed pseudo-timelike path around which the time direction reverses, preventing any consistent distinction between future- and past-directed vectors. To an observer near $\mathcal{H}$, such loops may appear as the creation of a particle-antiparticle pair at two distinct points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_02403 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Pseudo-timelike loops in signature changing semi-Riemannian manifolds with a transverse radical Rieger, N. E. Hasse, W. Differential Geometry General Relativity and Quantum Cosmology Mathematical Physics In 1983, Hartle and Hawking proposed the no-boundary proposal, suggesting that the universe has no beginning in the sense of a spacetime singularity or boundary. Nevertheless, there is an origin of time. Mathematically, this involves signature-type changing manifolds in which a Riemannian region smoothly transitions to a Lorentzian region across the hypersurface $\mathcal{H}$ where time begins. We develop a coherent framework for signature changing manifolds with a degenerate yet smooth metric. Established Lorentzian tools and results are then adapted to this setting, and new definitions are introduced that carry unforeseen causal implications. A noteworthy consequence is the presence of locally time-reversing loops through every point on the hypersurface. Imposing global hyperbolicity on the Lorentzian region, we prove that for every point $p \in M$ there exists a pseudo-timelike loop self-intersecting at $p$. Equivalently, $M$ always admits a closed pseudo-timelike path around which the time direction reverses, preventing any consistent distinction between future- and past-directed vectors. To an observer near $\mathcal{H}$, such loops may appear as the creation of a particle-antiparticle pair at two distinct points. |
| title | Pseudo-timelike loops in signature changing semi-Riemannian manifolds with a transverse radical |
| topic | Differential Geometry General Relativity and Quantum Cosmology Mathematical Physics |
| url | https://arxiv.org/abs/2409.02403 |