The representation of spacetime through time functions
Ettore Minguzzi
TL;DR
The paper proposes a minimal, time-centric model of spacetime in which topology, causal order, and a Lorentzian distance are encoded by time-like functions. It introduces a product trick that represents metric data as causality on an extended space, enabling a manifold-free (set-based) description. It proves a bidirectional correspondence: spacetime can be constructed from a family of rushing functions, and, under suitable topological conditions (e.g., $k_\omega$-spaces, local compactness, second-countability), every stable spacetime arises from continuous rushing functions. This framework unifies time and proper time and suggests that time fully characterizes spacetime, with potential implications for unifying gravity with quantum mechanics. The results generalize known manifold-based insights to abstract, non-smooth settings and provide constructive representations via function spaces.
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.