The weak equivalence principle states that freely falling test bodies follow universal trajectories independent of composition. Einstein elevated this to a geometrical insight: locally, gravity can be removed by choosing a freely falling frame, suggesting spacetime curvature as the true gravitational field.

General relativity replaces the Minkowski metric with a dynamical metric $g_{\mu\nu}(x)$. The Levi-Civita connection is $$\Gamma^\rho_{\mu\nu} = \frac{1}{2}g^{\rho\sigma}(\partial_\mu g_{\sigma\nu}+\partial_\nu g_{\sigma\mu}-\partial_\sigma g_{\mu\nu}).$$

The Riemann tensor $R^\rho{}_{\sigma\mu\nu}$ measures noncommutativity of covariant derivatives. Free-fall trajectories satisfy the geodesic equation $$\frac{d^2x^\mu}{d\lambda^2}+\Gamma^\mu_{\alpha\beta}\frac{dx^\alpha}{d\lambda}\frac{dx^\beta}{d\lambda}=0.$$

Einstein’s equations are $$G_{\mu\nu} + \Lambda g_{\mu\nu} = \frac{8\pi G}{c^4}T_{\mu\nu},$$ where $G_{\mu<br>u}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}$. They relate geometry to stress-energy.