Abstract
In this article we introduce a notion of normalized angle for Lorentzian pre-length spaces. This concept allows us to prove some equivalences to the definition of timelike curvature bounds from below for Lorentzian pre-length spaces. Specifically, we establish some comparison theorems known as the local Lorentzian version of the Toponogov theorem and the Alexandrov convexity property. Finally, as an application we obtain a first variation formula for non-negatively curved globally hyperbolic Lorentzian length spaces.
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Notes
Here, we set \({\mathcal {T}}(x,y)=0\) when the set of future-directed causal curves from x to y is empty.
Recall that \((X,d,\ll ,\le \tau )\) is causal if \(x\le y\) and \(y\le x\) imply \(x=y\).
Notice that comparison triangles are unique up to an isometry.
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Acknowledgements
The authors are very thankful to T. Beran for insightful comments on an earlier version of this work, to the organizers of SCRI21, a tribute to Roger Penrose for this outstanding event, and last but not least, to the anonymous referees, for their thorough work and helpful suggestions.
Funding
W. Barrera was partially supported by Conacyt under grants SNI 45382 and Ciencia de Frontera 21100. D. Solis was partially supported by Conacyt under grant SNI 38368. L. Montes de Oca acknowledges que support of Conacyt under de Becas Nacionales program (783177).
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Barrera, W., de Oca, L.M. & Solis, D.A. Comparison theorems for Lorentzian length spaces with lower timelike curvature bounds. Gen Relativ Gravit 54, 107 (2022). https://doi.org/10.1007/s10714-022-02989-2
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DOI: https://doi.org/10.1007/s10714-022-02989-2