Classification of solutions with isometries or homotheties

TL;DRAbstract

In specifying the symmetry properties of a metric one has to state the dimension of the maximal group of motions or homotheties, its algebraic structure, and the nature and dimension of its orbits. For this purpose we shall, as in §8.4, use the following notation: the symbols S, T and N will denote, respectively, spacelike, timelike and null orbits, and will be followed by a subscript giving the dimension. If an isometry group is transitive on the whole manifold V4, the space-time will be said to be homogeneous. If an isometry group is transitive on S3, T3 or N3, the spacetime will be called hypersurface-homogeneous (or, respectively, spatiallyhomogeneous, time-homogeneous, or null-homogeneous).

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