Geometric Relations between Relative Phase Difference and Hyperbolic Geometry in Special Relativity

Abstract

This study presents a geometric interpretation of key relations in special relativity through relative phase difference and phase unfolding. A relative phase difference Δθ is associated with the relative state of motion between two observers, connecting the circular relations sin Δθ and cos Δθ with the corresponding hyperbolic components sec Δθ and tan Δθ. Within this representation, relative velocity, proper-time ratio, Lorentz factor, spacetime components, and relativistic energy and momentum can be organized around the same relative phase relation. The framework is further extended to time-dependent motion through a relative phase path Δθ(t), including the accumulation of proper time in the twin paradox. The formulation preserves the standard mathematical relations of special relativity while providing a continuous geometric connection between circular phase geometry and hyperbolic geometry.