Solutions of the Einstein and Einstein–Maxwell equations for spherically symmetric metrics (those of Schwarzschild and Reissner–Nordstr\“{o}m) are derived and discussed in detail. The equations of orbits of planets and of bending of light rays in a weak field are derived and discussed. Two methods to measure the bending of rays are presented. Properties of gravitational lenses are described. The proof (by Kruskal) that the singularity of the Schwarzschild metric at r = 2m is spurious is given. The relation of the r = 2m surface to black holes is discussed. Embedding of the Schwarzschild spacetime in a 6-dimensional flat Riemann space is presented. The maximal extension of the Reissner–Nordstr\“{o}m metric (by the method of Brill, Graves and Carter) is derived. Motion of charged and uncharged particles in the Reissner–Nordstr\“{o}m spacetime is described.

This is an encyclopaedia of basic knowledge about the Kerr metric and related topics. It includes, among other things, the original Kerr derivation from Einstein’s equations via the Kerr–Schild metrics, the Carter derivation from the separability of the Klein–Gordon equation (a by-product thereof is the generalisation to nonzero cosmological constant), the derivation (with illustrations) of the formulae for the event horizons and stationary limit hypersurfaces, the derivation of Carter’s fourth first integral of geodesic equations, the discussion of properties of general geodesics and of geodesics in the equatorial plane, the maximal analytic extension by Boyer and Lindquist, the Penrose process of extracting angular momentum from a rotating black hole and the Bardeen proof of existence of locally nonrotating observers in a stationary-axisymmetric spacetime.