Causal Rigidity and the Single-Unit Universe: Integrating the Alexandrov-Zeeman and Unruh Clock Scales
Karl Svozil
TL;DR
The paper addresses how many fundamental dimensional constants are required in relativistic spacetime and argues that exactly one constant suffices. By integrating the operational view that a bona fide clock can derive spatial measurements from time with the geometric rigidity of causal structure (Alexandrov-Zeeman theorems), it shows that the light-cone structure fixes the metric up to a conformal factor, while a single clock fixes the global scale within that conformal class. The key result is a formal Uniqueness Theorem: the physical metric g lies in the conformal class [η] and can be written as g = λ^2 η, with λ determined unambiguously by clock readings, thereby selecting a unique metric from the conformal family. This unifies two perspectives and clarifies that the remaining constant is a single scale, with c serving as a kinematic unit conversion and the clock providing the necessary scale-breaking to render spacetime geometries physically distinct.
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.