From Active Lorentz Transformations to Covariant Dynamics with the Tetrad Formalism–A Didactical Approach

Lorentz transformations (LTs) are historically introduced as basis transformations in Special Rel-ativity (SR). However, their active interpretation physically describes a particle’s state by mappingits four-momentum before and after an interaction. Because active transformations induce suchphysical momentum changes, they can be directly used to describe the action of real forces. Sinceany descriptions of a law of nature, active LTs and their associated dynamics must be expressiblein a manifestly covariant form. But since standard curvilinear coordinate bases cannot supportlocal active Lorentz transformations in their standard form, this aforementioned covariance strictlyrequires the use of the tetrad formalism. While this foundational connection represents a brief butcrucial component of advanced relativity education, it is often overly condensed or omitted in stan-dard textbooks. This paper addresses this pedagogical limitation by providing a highly accessibleand structurally unified framework connecting SR and General Relativity (GR).By projecting standard Lorentz transformations with the tetrad fields, we obtain local,coordinate-dependent Lorentz transformations that preserve the spacetime metric at each point.From these active transformations, we formulate a general-covariant single-particle equation ofmotion (EOM) that unifies the standard geodesic equation and the Lorentz force law as specialcases.Supported by explicit examples, visualizations and direct comparisons, this work serves as atransparent instructional resource intended for instructors interested in presenting the tetrad for-malism and its applications and for advanced students seeking a deeper conceptual understandingof these relations.