Exploring the Mathematics of Spacetime in Einstein's Relativity

This paper maps the mathematical architecture of General Relativity, stepping away from basic coordinate-based tensor calculus to look at the deeper differential geometry at play. We formalize spacetime as a pseudo-Riemannian manifold $(M, g) that relies on a torsion-free Levi-Civita connection. By applying the principle of least action to the Einstein-Hilbert action, we derive the Einstein Field Equations from first principles. From there, we introduce Élie Cartan’s tetrad formalism. This approach proves mathematically indispensable when trying to couple fermionic fields to the geometry of spacetime. Ultimately, this framework shows that gravity isn't a traditional force at all. It is an intrinsic geometric deformation of the manifold itself, a reality that naturally paves the way for advanced extensions like Einstein-Cartan theory