Gloeckle, Jonathan (2024) An enlargeability obstruction for spacetimes with both big bang and big crunch. MATHEMATICAL RESEARCH LETTERS, 31 (5). pp. 1435-1469. ISSN 1073-2780, 1945-001X
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Given a spacelike hypersurface M of a time-oriented Lorentzian manifold ((M) over bar, (g) over bar), the pair (g, k) consisting of the induced Riemannian metric g and the second fundamental form k is known as initial data set. In this article, we study the space of all initial data sets (g, k) on a fixed closed n-manifold M that are subject to a strict version of the dominant energy condition. In this space we characterize big bang and big crunch initial data by tr k > 0 and tr k < 0, respectively. It is easy to see that these belong to the same path-component when M admits a positive scalar curvature metric. Conversely, it was observed in a previous work by the author [12] that this is not the case when the existence of a positive scalar curvature metric on M is obstructed in terms of the index in KO-n ({*}). In the present article we extend this disconnectedness result to Gromov-Lawson's enlargeability obstruction. In particular, for orientable closed 3-manifolds M, we can tell precisely when big bang and big crunch initial data belong to the same path-component. In the context of general relativity theory, this result may be interpreted as excluding the existence of certain globally hyperbolic spacetimes with both a big bang and a big crunch singularity.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | CURVATURE; 3-MANIFOLDS; METRICS; PROOF; SPIN; |
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 23 Jul 2025 05:04 |
| Last Modified: | 23 Jul 2025 05:04 |
| URI: | https://pred.uni-regensburg.de/id/eprint/63485 |
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