Observables for spacetimes with two Killing field symmetries

TL;DR

This paper analyzes spacetimes with two spacelike Killing symmetries using Hamiltonian methods, deriving an infinite set of classical observables related to integrable models and solution-generating techniques.

Contribution

It introduces a Hamiltonian framework for these spacetimes, explicitly constructs two infinite sets of observables, and connects them to integrable models and Geroch's solution-generating methods.

Findings

Derived infinite sets of classical observables as functionals of phase space variables.

Established a connection between observables and integrable model techniques.

Linked the observables to solution-generating methods like Geroch's technique.

Abstract

The Einstein equations for spacetimes with two commuting spacelike Killing field symmetries are studied from a Hamiltonian point of view. The complexified Ashtekar canonical variables are used, and the symmetry reduction is performed directly in the Hamiltonian theory. The reduced system corresponds to the field equations of the SL(2,R) chiral model with additional constraints. On the classical phase space, a method of obtaining an infinite number of constants of the motion, or observables, is given. The procedure involves writing the Hamiltonian evolution equations as a single `zero curvature' equation, and then employing techniques used in the study of two dimensional integrable models. Two infinite sets of observables are obtained explicitly as functionals of the phase space variables. One set carries sl(2,R) Lie algebra indices and forms an infinite dimensional Poisson algebra, while the other is formed from traces of SL(2,R) holonomies that commute with one another. The restriction of the (complex) observables to the Euclidean and Lorentzian sectors is discussed. It is also shown that the sl(2,R) observables can be associated with a solution generating technique which is linked to that given by Geroch.