Gravitation is induced by the stress-energy tensor of matter and fields via curvature. This four-covariant description contains the Newtonian limit of weak gravity and slow motion. Subject to conservation of energy and momentum, this leads uniquely to the Einstein equations of motion, up to a cosmological constant. These equations admit a Lagrangian by the associated scalar curvature, as described by the Hilbert action.

Curvature of spacetime displays features similar to that of the sphere, as in the previous chapter. It generalizes to four-dimensional spacetime as in the discussion of the gravitational field of a star.

Spacetime curvature is described by the Riemann tensor. Given a metric, and so the light cones at every point of spacetime, the Riemann tensor is defined completely by the metric up to its second coordinate derivatives. Both the Riemann tensor and the metric, each in different ways, contain time-independent gravitational interactions, including the Newtonian limit of weak gravity, as well as gravitational radiation.

The Riemann tensor has various representations which bring about different aspects of spacetime.

Parallel transport over a closed loop. Continuing the discussion of parallel transport on the sphere, consider vectors carried along closed curves in spacetime.