The existence of a timelike Killing vector is a strong restriction on solutions of the Einstein equations. It corresponds physically to an equilibrium situation. This case is considered in Section 3.1. A null Killing vector also represents a strong restriction. It seems to fit badly with the Cauchy problem and is thus far from the discussion in this paper. This case will not be discussed further here. There remains the case where all Killing vectors are spacelike. The classes of solutions can be distinguished by the dimension of the group orbits. The case where the orbits are three-dimensional is that of spatially homogeneous spacetimes, discussed in Section 3.2. If the group orbits are two-dimensional there are important classes of solutions where the group is three- or two-dimensional. The case of a three-dimensional group includes asymptotically flat spherically symmetric solutions, discussed in Sections 3.3 and 3.4. It also includes spatially compact spacetimes with spherical, plane and hyperbolic symmetry, for which results are included in Section 3.6. That section also covers spatially compact solutions with a two-dimensional isometry group. Most results concern the situation where the Killing vectors have no zeroes. This means that when the spacetime is quotiented by the symmetry group in order to get an effective lower-dimensional system, no singularities occur in the reduced equations of motion. This is a big advantage for analytical work. The assumption of exactly two spacelike Killing vectors is not compatible with asymptotic flatness. It does allow asymptotic flatness in all directions except one and this is the subject of Section 3.5.
A single Killing vector without fixed points gives rise to an interesting class of cosmological models. Because of the perceived closeness to solutions without any Killing vectors, the results on these spacetimes have been placed in Section 5.4. Spacetimes with one fixed-point free Killing vector which are partly asymptotically flat are mentioned briefly in Section 3.5. Axially symmetric asymptotically flat spacetimes have one spacelike Killing vector with zeroes on the axis. It seems that no analytical advantage over the general case has been found for these spacetimes and so there are no results to report.
© Max Planck Society and the author(s)