Derivation of the Lorentz Transformations for Three-Dimensional Relative Motion Between Inertial Frames of Reference

Lorentz transformations are central to relativistic mechanics, explaining phenomena such as the mass–energy relationship, spatial contraction, momentum transformation, velocity addition, and the geometric structure of Minkowski space-time. While traditionally derived for motion along a single spatial axis, these transformations can be generalized to three-dimensional space for greater applicability. In this article, we derive the Lorentz transformation equations for inertial frames in arbitrary three-dimensional relative motion. The resulting formalism provides a more symmetric and generalized framework, facilitating deeper insight into relativistic kinematics. Additionally, we introduce the concept of a position six-vector, comprising three spatial and three temporal components, which we employ to reformulate the Lorentz transformations, the d’Alembert operator, and the charge continuity equation within a six-dimensional space-time framework.