Some restrictions on the symmetry groups of axially symmetric spacetimes

Abstract

Lie transformation groups containing a one-dimensional subgroup acting cyclically on a manifold are considered. The structure of the group is found to be considerably restricted by the existence of a one-dimensional subgroup whose orbits are circles. The results proved do not depend on the dimension of the manifold nor on the existence of a metric, but merely on the fact that the Lie group acts globally on the manifold. Firstly, some results for the general case of an (m + 1)-dimensional Lie group are derived: those commutators of the associated Lie algebra involving the generator of the cyclic subgroup, X0 say, are severely restricted and, in a suitably chosen basis, take a simple form. The Jacobi identities involving X0 are then applied to show that there are further restrictions on the structure of the Lie algebra. All Lie algebras of dimensions two and three compatible with cyclic symmetry are obtained. In the two-dimensional case the group must be Abelian. For the three-dimensional case, the Bianchi type of the Lie algebra must be I, II, III, VII0, VIII or IX and furthermore in all cases the vector X0 forms part of a basis in which the algebra takes its canonical form. Finally, four-dimensional Lie algebras compatible with cyclic symmetry are considered and the results are related to the Petrov–Kruchkovich classification of all four-dimensional Lie algebras.