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The splitting theorem for globally hyperbolic Lorentzian length spaces with non-negative timelike curvature

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Abstract

In this work, we prove a synthetic splitting theorem for globally hyperbolic Lorentzian length spaces with global non-negative timelike curvature containing a complete timelike line. Inspired by the proof for smooth spacetimes (Beem et al. in Global differential geometry and global analysis 1984, Springer, pp. 1–13, 1985), we construct complete, timelike asymptotes which, via triangle comparison, can be shown to fit together to give timelike lines. To get a control on their behaviour, we introduce the notion of parallelity of timelike lines in the spirit of the splitting theorem for Alexandrov spaces as proven in Burago et al. (A course in metric geometry, vol 33, American Mathematical Society, Providence, 2001) and show that asymptotic lines are all parallel. This helps to establish a splitting of a neighbourhood of the given line. We then show that this neighbourhood has the timelike completeness property and is hence inextendible by a result in Grant et al. (Ann Glob Anal Geom 55(1):133–147, 2019), which globalises the local result.

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Notes

  1. Refer to chapter 14 in [9] for a detailed account.

  2. Geodesics defined on intervals [a, b] with \(a = -\infty \) or \(b=\infty \) should be thought of as properly reparametrised.

  3. A set \(V \subset X\) is called causally convex if for all \(p,q \in V\) we have \(J(p,q) \subset V\).

  4. If \(p\ll y_1\ll y_2\ll y_3\), we use Lemma 2.43 and if \(y_1\ll p\ll y_2\ll y_3\) we use Lemma 2.42.

  5. Approximating and globally causally closed Lorentzian pre-length spaces satisfy this, cf. [19, Def. 2.17].

  6. This agrees with the usual notion of synchronising clocks in the sense that this is the parametrisation coming from synchronising the clocks on the particles \(\alpha \) and \(\beta \) and parametrising \(\alpha \) and \(\beta \) by their clock values.

  7. So \(c_n\) converges to the point \([\partial _t]\) on the limit sphere.

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Acknowledgements

Argam Ohanyan and Felix Rott were supported by project P 33594 of the Austrian Science Fund FWF. Argam Ohanyan was also supported by the ÖAW-DOC scholarship of the Austrian Academy of Sciences. Didier Solis acknowledges the support of Conacyt under grant SNI 38368 and UADY under program FMAT-PTA2022, as well as the hospitality of the University of Vienna, where parts of this work were conducted. Tobias Beran acknowledges the support of University of Vienna. The authors would like to thank the Erwin Schrödinger Institute for support and hospitality during the programme Non-regular Spacetime Geometry where part of this work was undertaken. The authors would like to thank Matteo Calisti, Gregory Galloway, Melanie Graf, Michael Kunzinger, Clemens Sämann, Benedict Schinnerl and Roland Steinbauer for helpful discussions and valuable feedback. The authors would like to acknowledge the thorough work of the reviewer, whose suggestions helped improving this article.

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  1. Department of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090, Wien, Austria

    Tobias Beran, Argam Ohanyan & Felix Rott

  2. Facultad de Matemáticas, Universidad Autónoma de Yucatán, Anillo Periférico, Tablaje, 13615, Mérida, Mexico

    Didier A. Solis

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Beran, T., Ohanyan, A., Rott, F. et al. The splitting theorem for globally hyperbolic Lorentzian length spaces with non-negative timelike curvature. Lett Math Phys 113, 48 (2023). https://doi.org/10.1007/s11005-023-01668-w

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