General relativity replaces gravitational force with spacetime curvature described by a metric tensor $g_{\mu\nu}$. The Levi-Civita connection defines covariant differentiation, and the Riemann tensor captures tidal effects and geodesic deviation. Gravity becomes the geometry of spacetime itself.

The Einstein-Hilbert action is $S = \frac{1}{16\pi G}\int d^4x\,\sqrt{-g}\,R + S_{\text{matter}}$. Varying with respect to $g_{\mu\nu}$ yields Einstein's equations $G_{\mu\nu} = 8\pi G\,T_{\mu\nu}$. The action formulation makes diffeomorphism invariance and coupling to matter manifest.

Freely falling particles follow geodesics determined by the spacetime connection. Covariant conservation $\nabla_\mu T^{\mu\nu}=0$ reflects diffeomorphism invariance and the contracted Bianchi identity, linking local stress-energy conservation to geometry.

Black hole solutions reveal horizons, thermodynamic behavior, and quantum puzzles such as information loss. Cosmological solutions describe expanding universes, with the Friedmann equations emerging from symmetry-reduced Einstein dynamics. These topics connect gravitation with thermodynamics, quantum theory, and astrophysical observation.