A note on null distance and causality encoding

TL;DR

This paper examines the null distance in spacetime, showing that causality encoding can be achieved under weaker assumptions than previously thought, which broadens its applicability in mathematical physics.

Contribution

It demonstrates that the causality assumptions for null distance encoding can be relaxed, extending previous results to more general conditions.

Findings

Causality assumptions can be weakened in null distance encoding.

Null distance remains a useful metric for spacetime causality.

Broader conditions still allow for global causality encoding.

Abstract

Under natural conditions, the null distance introduced by Sormani and Vega [10] is a metric space distance function on spacetime, which, in a certain precise sense, can encode the causality of spacetime. The null distance function requires the choice of a time function. The purpose of this note is to observe that the causality assumptions related to such a choice in results used to establish global encoding of causality, due to Sakovich and Sormani [9] and to Burtscher and Garc\'ia-Heveling [2], can be weakened.