Einstein Equations with 1 Parameter Spacelike Isometry Group

Abstract

In a recent series of interesting articles V. Moncrief has obtained reduced hamiltonians for vacuum Einstein, or Einstein-Maxwell-Higgs equations. We shall in this paper show how one can obtain by direct methods, without making use of a hamiltonian formalism, the splitting of vacuum Einstein equations into a system elliptic on each spacelike slice on the one hand, and on the other hand a hyperbolic system which is an harmonic map equation from a pseudoriemannian manifold, whose metric depends on the solution of the elliptic system, into a fixed 2-dimensional symmetric riemannian space. We discuss, following with some variants ideas of Cameron and Moncrief, the general solution of the system, depending on the topology of space.