The representation of spacetime through time functions

TL;DR

This paper proposes a minimalistic, abstract representation of spacetime using families of time functions, demonstrating that time functions alone can fully characterize the structure of spacetime.

Contribution

It introduces a novel approach to represent spacetime solely through time functions, unifying causality and metric concepts in a minimal framework.

Findings

Spacetime can be represented as a family of functions over an arbitrary set.

Time functions fully characterize the topology, causality, and Lorentzian distance of spacetime.

The product trick unifies causality and metricity through an additional dimension.

Abstract

The properties of the stable distance over stable spacetimes are used as a reference to propose a simplified, abstract notion of spacetime. The discussion shows that spacetime, with its topology, causal order and (upper semi-continuous) Lorentzian distance, can be introduced in a general and minimalistic way. Specifically, it is shown that spacetime can be represented as nothing more than a family of functions defined over an arbitrary set, the functions being a posteriori interpreted as rushing time functions. The proof makes use of the product trick which reduces causality and metricity to causality in a space with one additional dimension, so leading to a kind of unification for the notions of time function and proper time. Ultimately, our results show that time fully characterizes spacetime.