The problem

When we are handling physical problems, symmetric systems have not only the advantage of a certain simplicity, or even beauty, but also special physical effects frequently occur then. One can therefore expect in General Relativity, too, that when a high degree of symmetry is present the field equations are easier to solve and that the resulting solutions possess special properties.

Our first problem is to define what we mean by a symmetry of a Riemannian space. The mere impression of simplicity which a metric might give is not of course on its own sufficient; thus, for example, the relatively complicated metric (31.1) in fact has more symmetries than the ‘simple’ plane wave (29.39). Rather, we must define a symmetry in a manner independent of the coordinate system. Here we shall restrict ourselves to continuous symmetries, ignoring discrete symmetry operations (for example, space reflections).

Killing vectors

The symmetry of a system in Minkowski space or in three-dimensional (Euclidean) space is expressed through the fact that under translation along certain lines or over certain surfaces (spherical surfaces, for example, in the case of spherical symmetry) the physical variables do not change. One can carry over this intuitive idea to Riemannian spaces and ascribe a symmetry to the space if there exists an s-dimensional (1 ≤ s ≤ 4) manifold of points which are physically equivalent: under a symmetry operation, that is, a motion which takes these points into one another, the metric does not change.