The future is not always open

• Published in: Letters in mathematical physics Vol. 110; no. 1; pp. 83 - 103
• Main Authors: Grant, James D. E., Kunzinger, Michael, Sämann, Clemens, Steinbauer, Roland
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01-01-2020; Springer Nature B.V
• Subjects: Causality; Complex Systems; Deformation mechanisms; Geometry; Group Theory and Generalizations; Mathematical and Computational Physics; Physics; Physics and Astronomy; Quantum gravity; Regularity; Theoretical; Low regularity; Causality theory; 53C50; 83C75; Chronological future; Causal bubbles
• ISSN: 0377-9017; 1573-0530
• DOI: 10.1007/s11005-019-01213-8

We demonstrate the breakdown of several fundamentals of Lorentzian causality theory in low regularity. Most notably, chronological futures (defined naturally using locally Lipschitz curves) may be non-open and may differ from the corresponding sets defined via piecewise
C
1
-curves. By refining the notion of a causal bubble from Chruściel and Grant (Class Quantum Gravity 29(14):145001, 2012), we characterize spacetimes for which such phenomena can occur, and also relate these to the possibility of deforming causal curves of positive length into timelike curves (
push-up
). The phenomena described here are, in particular, relevant for recent synthetic approaches to low-regularity Lorentzian geometry where, in the absence of a differentiable structure, causality has to be based on locally Lipschitz curves.