General Relativity
Curvature, geodesics, and Einstein's equations.
Spacetime is a 4-manifold with metric $g_{\mu\nu}$. The Levi-Civita connection $\Gamma^\lambda_{\mu\nu}$ is metric-compatible and torsion-free. Curvature: Riemann $R^\rho_{\ \sigma\mu\nu}$, Ricci $R_{\mu\nu}$, scalar $R$. Einstein's field equations:
$$G_{\mu\nu} = R_{\mu\nu} - \tfrac{1}{2}g_{\mu\nu}R = \frac{8\pi G}{c^4} T_{\mu\nu}.$$Free particles follow geodesics $\ddot x^\mu + \Gamma^\mu_{\alpha\beta}\dot x^\alpha \dot x^\beta = 0$. The Bianchi identity $\nabla^\mu G_{\mu\nu} = 0$ forces $\nabla^\mu T_{\mu\nu} = 0$.
Interactive: light bending near a mass
Photon trajectory through Schwarzschild geometry. Move the mass slider to see deflection grow.