Is time the real line?

TL;DR

This paper investigates the topological and algebraic structure of the arrow of time, proposing it as a normed vector space akin to the rational numbers and extending it to the real line, revealing a non-Hausdorff topology.

Contribution

It introduces a novel operational approach to characterize the arrow of time as a normed vector space and explores its topological properties within space-time.

Findings

The arrow of time can be modeled as a normed vector space $(\\mathbb{Q}, |\\cdot|)$.

Space-time is a fibration with sets of simultaneous events as fibers.

Classical space-time exhibits a non-Hausdorff topology.

Abstract

This paper is devoted to discussing the topological structure of the arrow of time. In the literature, it is often accepted that its algebraic and topological structures are that of a one-dimensional Euclidean space $\mathbb{E}^1$, although a critical review on the subject is not easy to be found. Hence, leveraging on an operational approach, we collect evidences to identify it structurally as a normed vector space $(\mathbb{Q}, |\cdot|)$, and take a leap of abstraction to complete it, up to isometries, to the real line. During the development of the paper, the space-time is recognized as a fibration, with the fibers being the sets of simultaneous events. The corresponding topology is also exposed: open sets naturally arise within our construction, showing that the classical space-time is non-Hausdorff. The transition from relativistic to classical regimes is explored too.