A Brief Introduction to General Relativity

This review provides a pedagogical introduction to the geometric foundations of general relativity and the origin of spacetime symmetries. Beginning with the concepts of topological and differentiable manifolds, we introduce local coordinate charts, transition maps, and atlases, emphasizing their role in endowing spacetime with a smooth differential structure. We then develop the notion of tangent spaces and vector fields, showing how local coordinate transformations induce changes of basis and naturally lead to the action of the general linear group GL(n). Interpreting these local frame transformations as a gauge symmetry highlights the close connection between diffeomorphism invariance and gauge invariance in general relativity. Building on these geometric foundations, we introduce the Lie derivative as the natural tool for comparing tensor fields along the flow of a vector field. This framework allows us to characterize spacetime symmetries through Killing vector fields, whose flows leave the metric invariant. As an explicit example, we derive the Killing vectors of Minkowski spacetime and show how they generate the translations, rotations, and Lorentz boosts that together form the Poincaré group. The article is intended as a concise and self-contained review for readers seeking an intuitive introduction to the differential-geometric language underlying modern gravitational physics.