The Petrov and Kaigorodov–Ozsváth solutions : spacetime as a group manifold

The Petrov solution (for Λ = 0) and the Kaigorodov–Ozsváth solution (for Λ < 0) provide examples of vacuum solutions of the Einstein equations with simply-transitive isometry groups. We calculate the boundary stress tensor for the Kaigorodov–Ozsváth solution in the context of the adS/CFT correspondence. By giving a matrix representation of the Killing algebra of the Petrov solution, we determine left-invariant 1-forms on the group. The algebra is shown to admit a two-parameter family of linear deformations a special case of which gives the algebra of the Kaigorodov–Ozsváth solution. By applying the method of nonlinear realizations to both algebras, we obtain a Lagrangian of Finsler type from the general first-order action in both cases. Interpreting the Petrov solution as the exterior solution of a rigidly rotating dust cylinder, we discuss the question of creation of CTCs by spinning up such a cylinder. We show geodesic completeness of the Petrov and Kaigorodov–Ozsváth solutions and determine the behaviour of geodesics in these spacetimes. The holonomy groups are shown to be given by the Lorentz group in both cases.