Einstein Tensor and Spherical Symmetry

The classification of symmetric second-rank tensors in Minkowski space and its application to the Einstein tensor is reviewed. It is shown that, for spherically symmetric metrics, the Einstein tensor always has a spacelike double eigenvector; and the possible types of Einstein tensor that this degeneracy allows are discussed. A complete classification of all spherically symmetric metrics with two double eigenvalues is given. A study of the timelike eigencongruence, in the case when one timelike and two spacelike eigenvectors exist, is carried out. Canonical forms for the metric, the Einstein tensor, and the Weyl tensor (which is always of type D) are given for each of the various possible types.