Symmetry preserving deformations of generalized Taub–NUT space-times to all orders in perturbation theory

Abstract

Generalized Taub–NUT (Newman–Unti–Tamburino) space-times have compact Cauchy horizons and (generically) admit one (spacelike) Killing field in their globally hyperbolic regions. A large family of such vacuum space-times can be defined on circle bundles over K×R, where K is a compact two-manifold, with the circular fibers of the bundle being defined by the orbits of the Killing field. For the simplest case of product circle bundles the symmetry preserving vacuum perturbations of such backgrounds to arbitrarily high order in perturbation theory are considered. The analytic form of the general solution of the nth-order perturbation equations for all n is derived under the restriction that the perturbations considered preserve the (one Killing field) symmetry of the background. The evolution equations are treated first and then the constraint equations are imposed, recovering along the way the well-known result that linearization instabilities arise only if one attempts to perturb from one (Killing) symmetry class to another. Gauge transformations, decompositions, and the natural symplectic structure associated with the perturbation formalism are also discussed. The possibility of extending these results to the case of symmetry breaking perturbations and of using the results to derive the asymptotic behavior of solutions near their singular boundaries is briefly discussed.