Characterization of some causality conditions through the continuity of the Lorentzian distance

Abstract

A classical result in Lorentzian geometry states that a strongly causal spacetime is globally hyperbolic if and only if the Lorentzian distance is finite valued for every metric choice in the conformal class. It is proven here that a non-total imprisoning spacetime is globally hyperbolic if and only if for every metric choice in the conformal class the Lorentzian distance is continuous. Moreover, it is proven that a non-total imprisoning spacetime is causally simple if and only if for every metric choice in the conformal class the Lorentzian distance is continuous wherever it vanishes. Finally, a strongly causal spacetime is causally continuous if and only if there is at least one metric in the conformal class such that the Lorentzian distance is continuous wherever it vanishes.Comment: 14 pages, 2 figure. v2: Added material on global hyperbolicity. The title has changed. Previous title: Characterization of causal simplicity and causal continuity through the continuity of the Lorentzian distance. v3: Some misprints fixed. Final versio