A lower semicontinuous time separation function for $C^0$ spacetimes

TL;DR

This paper introduces a new class of curves called 'nearly timelike' to define a lower semicontinuous time separation function in $C^0$ spacetimes, extending properties known in smooth cases and ensuring better geometric analysis.

Contribution

It defines nearly timelike curves and shows the associated time separation function is lower semicontinuous in $C^0$ spacetimes, aligning with the smooth case and providing conditions for maximizers.

Findings

The new time separation function is lower semicontinuous for $C^0$ spacetimes.

It coincides with the classical function in smooth spacetimes.

Conditions for the existence of nearly timelike maximizers are established.

Abstract

The time separation function (or Lorentzian distance function) is a fundamental object used in Lorentzian geometry. For smooth spacetimes it is known to be lower semicontinuous, and in fact, continuous for globally hyperbolic spacetimes. Moreover, an axiom for Lorentzian length spaces - a synthetic approach to Lorentzian geometry - is the existence of a lower semicontinuous time separation function. Nevertheless, the usual time separation function is $\textit{not}$ necessarily lower semicontinuous for $C^0$ spacetimes due to bubbling phenomena. In this paper, we introduce a class of curves called "nearly timelike" and show that the time separation function for $C^0$ spacetimes is lower semicontinuous when defined with respect to nearly timelike curves. Moreover, this time separation function agrees with the usual one when the metric is smooth. Lastly, sufficient conditions are found guaranteeing the existence of a nearly timelike maximizer between two points in a $C^0$ spacetime.