Abstract:The study of low regularity (in-)extendibility of Lorentzian manifolds is motivated by the question whether a given solution to the Einstein equations can be extended (or is maximal) as a weak solution. In this paper we show that a timelike complete and globally hyperbolic Lorentzian manifold is C 0 -inextendible. For the proof we make use of the result, recently established by Sämann (Ann Henri Poincaré 17(6): 2016), that even for continuous Lorentzian manifolds that are globally hyperbolic, there exists a length-maximizing causal curve between any two causally related points.
The existence, established over the past number of years and supporting earlier work of Ori [14], of physically relevant black hole spacetimes that admit C 0 metric extensions beyond the future Cauchy horizon, while being C 2inextendible, has focused attention on fundamental issues concerning the strong cosmic censorship conjecture. These issues were recently discussed in the work of Jan Sbierski [17], in which he established the (nonobvious) fact that the Schwarschild solution in global Kruskal-Szekeres coordinates is C 0 -inextendible. In this paper we review aspects of Sbierski's methodology in a general context, and use similar techniques, along with some new observations, to consider the C 0 -inextendibility of open FLRW cosmological models. We find that a certain special class of open FLRW spacetimes, which we have dubbed 'Milne-like,' actually admit C 0 extensions through the big bang. For spacetimes that are not Milne-like, we prove some inextendibility results within the class of spherically symmetric spacetimes.
In this note we present a result establishing the existence of a compact CMC Cauchy surface from a curvature condition related to the strong energy condition.
We prove that causal maximizers in C 0,1 spacetimes are either timelike or null. This question was posed in [17] since bubbling regions in C 0,α spacetimes (α < 1) can produce causal maximizers that contain a segment which is timelike and a segment which is null, cf. [3]. While C 0,1 spacetimes do not produce bubbling regions, the causal character of maximizers for spacetimes with regularity at least C 0,1 but less than C 1,1 was unknown until now. As an application we show that timelike geodesically complete spacetimes are C 0,1 -inextendible. *
This paper serves as an introduction to $$C^0$$
C
0
causal theory. We focus on those parts of the theory which have proven useful for establishing spacetime inextendibility results in low regularity—a question which is motivated by the strong cosmic censorship conjecture in general relativity. This paper is self-contained; prior knowledge of causal theory is not assumed.
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