Hyperbolic Angles in Lorentzian Space: A Physical and Geometric Perspective

Abstract

In this study, we investigate hyperbolic angles in the two- dimensional Lorentzian plane L² equipped with the metric signature (+,-), from both geometric and physical perspectives. Using light signal--based constructions and Bondi factors, explicit formulas are derived for the hyperbolic angles formed by unit and non-unit timelike and spacelike vectors with the coordinate axes. These formulas are further interpreted within the geometric framework of special relativity. We also show that hyperbolic angles are completely determined by logarithmic measurements along the time axis, regardless of whether the underlying vectors are timelike or spacelike. Furthermore, by introducing suitably defined Euclidean-Lorentz and Lorentz- Euclidean circular projection mappings, we establish a bijective correspondence between hyperbolic angles in Lorentzian plane and Euclidean angles in the Euclidean plane. In particular, explicit relations between Euclidean and Lorentzian rotations are derived, leading to the identity α=arctan(tanhβ), where α denotes the Euclidean angle and β denotes the hyperbolic angle. The obtained results provide a unified geometric interpretation of Euclidean and hyperbolic angle structures and clarify the relationship between Euclidean and Lorentzian rotations in the Lorentzian plane.