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.

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.