It is shown how the assumption of symmetry implies the Killing equations (more generally, invariance equations of arbitrary tensors are derived and discussed). It is also shown how to find the symmetry transformations of a manifold given the Killing vectors. The Lie derivative is introduced, and it is shown that the algebra of a symmetry group always has a finite dimension, not larger than n(n+1)/2, where nis the dimension of the manifold. Conformal symmetries are defined and it is shown that the algebra of the conformal symmetry group has dimension not larger than (n+1)(n+2)/2. The metric of a spherically symmetric 4-dimensional manifold is derived from the Killing equations, and its general properties are discussed. Explicit formulae for the conformal symmetries of a flat space of arbitrary dimension are given.

Spherical symmetry for a metric is defined and used to build a two-function ansatz. The Schwarzschild spacetime emerges as the solution to Einstein’s equation with, and we see how the Newtonian potential is related to the linearized Schwarzschild metric. The lightlike and spacelike geodesics of the metric are explored using exact, approximate, and numerical approaches. Many of the usual experimental tests are covered in detail: perihelion precession, bending of light, and time dilation, for example. The structure of the singularities in the Schwarzschild spacetime is studied using Eddington–Finkelstein and Kruskal–Szekeres coordinates. At the end of the chapter, the Kerr spacetime is introduced, and students are invited to explore its geodesics.