Causality theory for closed cone structures with applications

Abstract

We develop causality theory for upper semi-continuous distributions of cones over manifolds generalizing results from mathematical relativity in two directions: non-round cones and non-regular differentiability assumptions. We prove the validity of most results of the regular Lorentzian causality theory including causal ladder, Fermat's principle, notable singularity theorems in their causal formulation, Avez-Seifert theorem, characterizations of stable causality and global hyperbolicity by means of (smooth) time functions. For instance, we give the first proof for these structures of the equivalence between stable causality, $K$-causality and existence of a time function. The result implies that closed cone structures that admit continuous increasing functions also admit smooth ones. We also study proper cone structures, the fiber bundle analog of proper cones. For them we obtain most results on domains of dependence. Moreover, we prove that horismos and Cauchy horizons are generated by lightlike geodesics, the latter being defined through the achronality property. Causal geodesics and steep temporal functions are obtained with a powerful product trick. The paper also contains a study of Lorentz-Minkowski spaces under very weak regularity conditions. Finally, we introduce the concepts of stable distance and stable spacetime solving two well known problems (a) the characterization of Lorentzian manifolds embeddable in Minkowski spacetime, they turn out to be the stable spacetimes, (b) the proof that topology, order and distance (with a formula a la Connes) can be represented by the smooth steep temporal functions. The paper is self-contained, in fact we do not use any advanced result from mathematical relativity.

Latex2e, 138 pages. Work presented at the meetings "Non-regular spacetime geometry", Firenze, June 20-22, 2017, and "Advances in General Relativity", ESI Vienna, August 28 - September 1, 2017. v3: added distance formula for stably causal (rather than just stable) spacetimes. v4: Added a few regularity results, final version

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• The future is not always open
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• The splitting theorem for globally hyperbolic Lorentzian length spaces with non-negative timelike curvature
• The singularity theorems of General Relativity and their low regularity extensions
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• A Lorentzian analog for Hausdorff dimension and measure
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• Remarks on the cosmological constant appearing as an initial condition for Milne-like spacetimes
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• On the asymptotic assumptions for Milne-like spacetimes
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• Further observations on the definition of global hyperbolicity under low regularity
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• Adiabatic Ground States in Non-Smooth Spacetimes
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