The Mathematical Framework of General Relativity

General Relativity is founded on the axiomatic assumption that space and time together form a four-dimensional pseudo-Riemannian manifold known as spacetime. Within this framework, curvature is encoded by a metric tensor and its associated geometric quantities. This paper develops the mathematical foundations underlying general relativity by introducing line elements, metric tensors, and transition maps on smooth manifolds, and by verifying and discussing elementary solutions to the Einstein Field Equations.