The metric tensor and the (pseudo-)Riemannian manifolds are defined. The results of the earlier chapters are specialised to this case, in particular the affine connection coefficients are shown to reduce to the Christoffel symbols. The signature of a metric, the timelike, null and spacelike vectors are defined and the notion of a light cone is introduced. It is shown that in two dimensions the notion of curvature agrees with intuition. It is also shown that geodesic lines extremise the interval (i.e. the ‘distance’). Mappings between Riemann spaces are discussed. Conformal curvature (= the Weyl tensor) is defined and it is shown that zero conformal curvature on a manifold of dimension >=4 implies that the metric is proportional to the flat one. Conformal flatness in three dimensions and the Cotton–York tensor are discussed. Embeddings of Riemannian manifolds in Riemannian manifolds of higher dimension are discussed and the Gauss–Codazzi equations derived. The Petrov classification of conformal curvature tensors in four dimensions with signature (+ - - -) is introduced at an elementary level.

Both experimentally and theoretically, the curved spacetimes of general relativity are explored by studying how test particles and light rays move through them. This chapter derives and analyzes the equations governing the motion of test particles and light rays in a general curved spacetime. Only test particles free from any influences other than the curvature of spacetime (electric forces, for instance) are considered. Such particles are called free, or freely falling, in general relativity. In general relativity, free means free from any influences besides the curvature of spacetime. We begin with the equations of motion for test particles with nonvanishing rest mass moving on timelike world lines, and revisit the equations of motion for light rays.