On the gauge fixing of 1 Killing field reductions of canonical gravity: the case of asymptotically flat induced 2-geometry

TL;DR

This paper investigates gauge fixing in 1 Killing field reductions of 4D vacuum gravity, focusing on asymptotically flat 2D geometries, and explores the Hamiltonian formulation and solution properties of the resulting equations.

Contribution

It provides a detailed analysis of gauge fixing and Hamiltonian reduction for 1 Killing field vacuum gravity with asymptotic flatness, including explicit solutions and PDE analysis.

Findings

Explicit solutions to diffeomorphism constraints

Relation of Hamiltonian constraint to negative curvature equations

Partial results on existence and uniqueness of solutions

Abstract

We consider 1 spacelike Killing vector field reductions of 4-d vacuum general relativity. We restrict attention to cases in which the manifold of orbits of the Killing field is $R^{3}$. The reduced Einstein equations are equivalent to those for Lorentzian 3-d gravity coupled to an SO(2,1) nonlinear sigma model on this manifold. We examine the theory in terms of a Hamiltonian formulation obtained via a 2+1 split of the 3-d manifold. We restrict attention to geometries which are asymptotically flat in a 2-d 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 2-d conformal flatness and maximal slicing. We explicitly solve the diffeomorphism constraints and relate the Hamiltonian constraint to the prescribed negative curvature equation in $R^2$ studied by mathematicians. We partially address issues of existence and/or uniqueness of solutions to the various elliptic partial differential equations encountered.