Globally hyperbolic spacetimes can be defined without the 'causal' condition

Abstract

Reasonable spacetimes are non-compact and of dimension larger than two. We show that these spacetimes are globally hyperbolic if and only if the causal diamonds are compact. That is, there is no need to impose the causality condition, as it can be deduced. We also improve the definition of global hyperbolicity for the non-regular theory (non $C^{1,1}$ metric) and for general cone structures by proving the following convenient characterization for upper semi-continuous cone distributions: causality and the causally convex hull of compact sets is compact. In this case the causality condition cannot be dropped, independently of the spacetime dimension. Similar results are obtained for causal simplicity.

arXiv commentLatex2e, 12 pages. v2: fixed a few typos

References in corpus (20)

• On smooth Cauchy hypersurfaces and Geroch's splitting theorem
• Globally hyperbolic spacetimes can be defined as "causal" instead of "strongly causal"
• Causality theory for closed cone structures with applications
• On Lorentzian causality with continuous metrics
• Convex neighborhoods for Lipschitz connections and sprays
• On the interplay between Lorentzian Causality and Finsler metrics of Randers type
• Lyapounov Functions of closed Cone Fields: from Conley Theory to Time Functions
• Global hyperbolicity for spacetimes with continuous metrics
• Timelike completeness as an obstruction to $C^0$-extensions
• A regularisation approach to causality theory for $C^{1,1}$-Lorentzian metrics
• Eisenhart's theorem and the causal simplicity of Eisenhart's spacetime
• On differentiability of volume time functions
• Characterization of some causality conditions through the continuity of the Lorentzian distance
• Even the Minkowski space is holed
• Causally simple inextendible spacetimes are hole-free
• Static- and Stationary-complete Spacetimes: Algebraic and Causal Structures
• Convexity and quasi-uniformizability of closed preordered spaces
• Conformally embedded spacetimes and the space of null geodesics
• On the causal properties of warped product spacetimes
• Conjectures on the Relations of Linking and Causality in Causally Simple Spacetimes

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• The singularity theorems of General Relativity and their low regularity extensions
• Aspects of $C^0$ causal theory
• Globally hyperbolic spacetimes: slicings, boundaries and counterexamples
• Global Hyperbolicity through the Eyes of the Null Distance
• On sliced spaces: Global Hyperbolicity revisited
• Further observations on the definition of global hyperbolicity under low regularity
• Inverse problems for non-linear hyperbolic equations with disjoint sources and receivers
• On Global Hyperbolicity of Spacetimes: Topology Meets Functional Analysis
• On the regularity of Cauchy hypersurfaces and temporal functions in closed cone structures
• Einstein causality of quantum measurements in the Tomonaga-Schwinger picture
• Causal ladder of Finsler spacetimes with a cone Killing vector field
• Global Hyperbolicity and Self-adjointness
• On Completeness of the Alexandrov Topology on a Spacetime: remarks and some corrections