Limit curve theorems in Lorentzian geometry

TL;DR

This paper reviews and generalizes limit curve theorems in Lorentzian geometry, exploring their applications to causality conditions and the structure of spacetime, including properties of lightlike lines and chronology violating sets.

Contribution

It formulates a comprehensive limit curve theorem encompassing various convergence scenarios and applies it to analyze causality and the structure of Lorentzian manifolds.

Findings

Strong causality is either everywhere true or false on maximizing lightlike segments.

If two lightlike lines intersect, strong causality holds at their points.

Disjoint closures of chronology violating set components or lightlike lines pass through their boundary intersections.

Abstract

The subject of limit curve theorems in Lorentzian geometry is reviewed. A general limit curve theorem is formulated which includes the case of converging curves with endpoints and the case in which the limit points assigned since the beginning are one, two or at most denumerable. Some applications are considered. It is proved that in chronological spacetimes, strong causality is either everywhere verified or everywhere violated on maximizing lightlike segments with open domain. As a consequence, if in a chronological spacetime two distinct lightlike lines intersect each other then strong causality holds at their points. Finally, it is proved that two distinct components of the chronology violating set have disjoint closures or there is a lightlike line passing through each point of the intersection of the corresponding boundaries.