Mathematics
Curvature in Riemannian Geometry and General Relativity
Quick fact
In general relativity, the presence of mass and energy literally tells spacetime how to curve, and that curvature then tells objects how to move. This geometric description of gravity perfectly explains phenomena like the bending of light around the Sun.
Why this is interesting
You've likely heard that gravity is the curvature of spacetime, but what does that really mean? How can empty space itself be bent?
Read the full explanation
Understanding Curvature in Riemannian Geometry and General Relativity
Think of a curved surface like a saddle or a sphere. You can measure its curvature locally by walking around on it, without ever leaving the surface. That's the idea of intrinsic curvature. Riemann generalized this from 2D surfaces to arbitrary dimensions. In general relativity, spacetime is a 4-dimensional manifold with a metric that determines distances. The curvature of this spacetime is what we experience as gravity.
A deeper explanation
The Riemann curvature tensor measures how the spacetime manifold bends. It is built from the metric and its derivatives and quantifies how parallel-transporting a vector around a small loop changes it. In 2D, the Gaussian curvature is a single number, but in 4D, curvature is a tensor with many components. The Ricci tensor is a contraction of the Riemann tensor, and it appears in Einstein's field equations, which relate the geometry of spacetime to the energy-momentum content. This makes gravity a purely geometric phenomenon: massive objects curve spacetime, and other objects follow geodesics, the straightest paths, in that curved spacetime.