A globally hyperbolic Lorentzian length space of the form Σ × R with compact Σ and non-negative timelike curvature splits as a metric Lorentzian product, provided its vertical curves are timelike complete and chronologically related.
Dimensions of ordered spaces and Lorentzian length spaces
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abstract
After calculating the Dushnik-Miller dimension of Minkowski spaces to be countable infinity, we define a novel notion of dimension for ordered spaces recovering the correct manifold dimension and obtain a corresponding obstruction for the existence of injective monotonous maps between Lorentzian length spaces. Furthermore we induce metrics on Cauchy subsets, relate respective Hausdorff dimensions, prove existence of rushing Cauchy functions with a given Cauchy zero locus and consider collapse phenomena in this setting.
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Low regularity approach to Bartnik's conjecture
A globally hyperbolic Lorentzian length space of the form Σ × R with compact Σ and non-negative timelike curvature splits as a metric Lorentzian product, provided its vertical curves are timelike complete and chronologically related.