Causal Rigidity and the Single-Unit Universe: Integrating the Alexandrov-Zeeman and Unruh Clock Scales

TL;DR

This paper unifies two perspectives on relativistic spacetime, demonstrating that a single fundamental constant suffices to determine spacetime geometry by linking conformal structure with clock normalization.

Contribution

It formally shows that a single clock normalization can select a unique spacetime metric from a conformal class, integrating operational and mathematical viewpoints.

Findings

Theorems establish the conformal structure of spacetime is rigid.

Clock normalization breaks dilation symmetry.

Number of fundamental constants is exactly one.

Abstract

We unify two complementary viewpoints on relativistic spacetime and the counting of fundamental constants. Operationally, Matsas, Pleitez, Saa, and Vanzella (MPSV) have recently argued that relativistic spacetime requires only a single fundamental dimensional constant. Mathematically, theorems due to Alexandrov and Zeeman demonstrate that the light-cone structure determines the spacetime geometry only up to a conformal factor. We show that these approaches are mutually reinforcing: the Alexandrov-Zeeman theorems establish the rigid conformal structure of spacetime, while the ``bona fide clock'' required by MPSV serves the necessary mathematical role of breaking the dilation symmetry. We provide a formal derivation proving that the normalization of a single clock worldline is sufficient to select a unique metric from the conformal class, thereby clarifying that the number of fundamental constants is exactly one.