On the Impossibility of Deriving Length Contraction from a Purely Horizontal Light Clock

This paper investigates whether the relativistic length contraction formula can be derived independently from Einstein’s postulates of special relativity using a horizontal light clock thought experiment. While time dilation is commonly derived using a vertical light clock, this work analyzes whether an analogous horizontal configuration can yield the length contraction relation without prior assumptions about time dilation. By comparing the round-trip light path in the rest frame of the rod and the laboratory frame, the derivation produces an expression L = L0/γ² when time intervals are assumed equal between frames. The discrepancy with the standard relativistic expression L = L0/γ is then examined and shown to arise from the implicit assumption that time passes identically in both frames. The analysis demonstrates that time dilation is inherently required for a consistent derivation of length contraction in this framework. In contrast, time dilation can be derived independently without first assuming length contraction. This reveals a pedagogical asymmetry in light-clock-based derivations and highlights the logically prior role of time dilation within the structure of special relativity.