The Petrov and Kaigorodov-Ozsv\'ath Solutions: Spacetime as a Group Manifold

TL;DR

This paper explores specific vacuum solutions to Einstein's equations, analyzing their algebraic structures, boundary stress-tensors, and geodesic properties, with implications for spacetime symmetries and the creation of closed timelike curves.

Contribution

It provides a detailed algebraic and geometric analysis of the Petrov and Kaigorodov-Ozsváth solutions, including their deformations, boundary stress-tensors, and geodesic completeness, linking spacetime symmetries to physical properties.

Findings

The algebra admits a two-parameter family of deformations.

The boundary stress-tensor is computed within the adS/CFT framework.

Both solutions are geodesically complete with Lorentz holonomy groups.

Abstract

The Petrov solution (for $\Lambda=0$) and the Kaigorodov-Ozsv\'ath solution (for $\Lambda<0$) provide examples of vacuum solutions of the Einstein equations with simply-transitive isometry groups. We calculate the boundary stress-tensor for the Kaigorodov-Ozsv\'ath 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 one-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-Ozsv\'ath solution. By applying the method of non-linear realisations 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-Ozsv\'ath solutions and determine the behaviour of geodesics in these spacetimes. The holonomy groups were shown to be given by the Lorentz group in both cases.