Reduction of Einstein’s equations for vacuum space-times with spacelike U(1) isometry groups

Abstract

We consider vacuum space-times with spacelike U(1) isometry groups which are defined on manifolds of the form R × Bn, where Bn is an arbitrary S1-bundle over the two-sphere. We reduce the Einstein equations for this problem to a system of “harmonic map” equations defined over the base manifold R × S2 equipped with a Lorentzian metric determined uniquely by the solution of an associated nonlinear elliptic system. The harmonic map fields (which have a range space diffeomorphic to the hyperbolic two-plane) represent the unconstrained, dynamical degrees of freedom of the gravitational field. We give a complete discussion of the existence and uniqueness of solutions of the associated elliptic system and also display a Geroch-type solution generating technique for globally transforming the space of solutions associated with any one non-trivial bundle, R × Bn → R × S2, to that of any other such bundle. The basic techniques could readily be generalized to treat S1-bundles over R × M where M is a compact two-manifold of arbitrary genus. In the higher genus cases the Teichmüller space of conformally diffeomorphic Riemannian metrics over M arises as an additional component of the configuration manifold. For M ≈ S2 this space collapses to a point, which slightly simplifies the analysis.