Abstract
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.
Highlights
Lorentzian causality theory had mostly been studied under the assumption of a smooth spacetime metric
While some features of causality theory are rather robust and topological in nature, other results are usually proved by local arguments involving geodesically convex neighbourhoods, which do exist for C1,1-metrics [22,28]
Arguments from causality theory which neither explicitly involve the exponential map nor geodesics have been found to extend to locally Lipschitz metrics
Summary
Lorentzian causality theory had mostly been studied under the assumption of a smooth spacetime metric. The exterior bubbling set is closely connected to push-up properties To relate these to the concepts introduced above, let us first give a formal definition: Definition 2.11 A continuous spacetime (M, g) is said to possess the push-up property if the following holds: Whenever γ : [a, b] → M is a (absolutely continuous) future/past-directed causal curve from p = γ (a) to q = γ (b) and if {t ∈ [a, b] :. (i)⇒(iii): By covering γ ([a, b]) with cylindrical charts U contained in the given neighbourhood of γ ([a, b]), it suffices to show push-up for a curve γ : [a, b] → U emanating from p that is future-directed causal and such that the set A := {t ∈ [a, b] : γ (t) exists and is timelike} has positive Lebesgue measure λ(A). We provide examples that answer all of these questions in the negative
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