Educational approach to general relativity through its approximate vectorial form

Abstract

General Relativity (GR) can be effectively examined in the low-velocity, weak-field limit, where its inherently tensorial structure simplifies to a vector-based formulation closely analogous to classical electrodynamics. This correspondence becomes particularly clear when both theories are expressed within a field-free, interaction-based framework, in which the leading relativistic corrections emerge naturally as velocity- and acceleration-dependent modifications to the static interaction. As a result, several predictions of GR can be derived using familiar techniques from electrodynamics, provided the relevant phenomena remain within the regime of validity of the approximation. This structural analogy facilitates a conceptual understanding of GR enabling the introduction of its core ideas even in an introductory level course, the single exception being the treatment of radiation, which requires tools at more advanced level. Building on this pedagogical framework, the incorporation of special relativity (SR) and the equivalence principle enables concise and transparent derivations of gravitational time dilation, as well as its immediate consequences, including light deflection and gravitational frequency shifts. A detailed exposition of this material is provided in the supplementary material.