Gauge fixing of one Killing field reductions of canonical gravity: The case of asymptotically flat induced two-geometry

Abstract

We consider one spacelike Killing vector field reductions of four-dimensional (4D) vacuum general relativity. We restrict attention to cases in which the manifold of the orbits of the Killing field is R3. The reduced Einstein equations are equivalent to those for Lorentzian 3D gravity coupled to an SO(2,1) nonlinear σ model on this manifold. We examine the theory in terms of a Hamiltonian formulation obtained via a 2+1 split of the 3D manifold. We restrict attention to geometries which are asymptotically flat in a 2D sense defined recently. We attempt to pass to a reduced Hamiltonian description in terms of the true degrees of freedom of the theory via gauge-fixing conditions of 2D conformal flatness and maximal slicing. We explicitly solve the diffeomorphism constraints and relate the Hamiltonian constraint to the prescribed negative curvature equation in R2 studied by mathematicians. We partially address issues of existence and/or uniqueness of solutions to the various elliptic partial differential equations encountered.