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.

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 version

References in corpus (6)

• Globally hyperbolic spacetimes can be defined as "causal" instead of "strongly causal"
• The causal hierarchy of spacetimes
• Limit curve theorems in Lorentzian geometry
• Non-imprisonment conditions on spacetime
• Weak distinction and the optimal definition of causal continuity
• On the causal properties of warped product spacetimes

Cited by in corpus (14)

• Causality theory for closed cone structures with applications
• Lyapounov Functions of closed Cone Fields: from Conley Theory to Time Functions
• Global hyperbolicity for spacetimes with continuous metrics
• On differentiability of volume time functions
• Globally hyperbolic spacetimes can be defined without the 'causal' condition
• Global hyperbolicity is stable in the interval topology
• The singularity theorems of General Relativity and their low regularity extensions
• Lorentzian Manifolds and Causal Sets as Partially Ordered Measure Spaces
• Causality of spacetimes admitting a parallel null vector and weak KAM theory
• Further observations on the definition of global hyperbolicity under low regularity
• Green Operators in Low Regularity Spacetimes and Quantum Field Theory
• Smoothing causal functions
• The global properties of the finiteness and continuity of the Lorentzian distance
• Spacetime closedness theorem for homogeneous Lorentzian foliations