The cosmological time functions and lightlike rays

TL;DR

This paper investigates the properties of cosmological time functions in space-time, showing how discontinuities relate to lightlike rays and providing conditions for regularity and causal properties.

Contribution

It establishes the relationship between discontinuities of cosmological time functions and lightlike rays, and offers new conditions for regularity and causality in space-time.

Findings

Discontinuities of cosmological time functions occur on past lightlike rays.

Regularity of the cosmological time function implies certain causal properties.

Conditions for causal simplicity and pseudoconvexity are reduced and clarified.

Abstract

It is proved that all discontinuity points of a finite cosmological time function, $\tau$, are on past lightlike rays. As a result, it is proved that if $(M,g)$ is a chronological space-time without past lightlike rays then there is a representation of $g$ such that its cosmological time function is regular. In addition, by reducing conditions of regularity sufficient conditions for causal simplicity and causal pseudoconvexity of space-time is given. It is also proved that the second condition of regularity can be reduced to satisfies only on inextendible past-directed causal rays if $(M,g)$ be a space-time, conformal with an open subspace of Minkowski space-time or $\tau$ be continuous.