Is time the real line?

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.

arXiv comment26 pages, 9 figures. Misprints corrected and minor adjustments according to the review process

References in corpus (5)

• On the possibility of experimental detection of the discreteness of time
• Geometric Justification of the Fundamental Interaction Fields for the Classical Long-Range Forces
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• General theory of non-reversible local dynamics
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Cited by in corpus (1)

• A Categorical View on the Principle of Relativity