Einstein's equations and the chiral model

TL;DR

This paper reformulates vacuum Einstein equations with two symmetries using Ashtekar variables, revealing a modified chiral model structure with explicit time-dependent observables that solve the Hamiltonian equations.

Contribution

It introduces a novel formulation of Einstein's equations as a time-dependent SL(2) chiral model, providing explicit solutions for invariant observables.

Findings

Reformulation as a modified SL(2) principal chiral model

Explicit time dependence of invariant observables derived

Solutions to Hamiltonian Einstein equations obtained

Abstract

The vacuum Einstein equations for spacetimes with two commuting spacelike Killing field symmetries are studied using the Ashtekar variables. The case of compact spacelike hypersurfaces which are three-tori is considered, and the determinant of the Killing two-torus metric is chosen as the time gauge. The Hamiltonian evolution equations in this gauge may be rewritten as those of a modified SL(2) principal chiral model with a time dependent `coupling constant', or equivalently, with time dependent SL(2) structure constants. The evolution equations have a generalized zero-curvature formulation. Using this form, the explicit time dependence of an infinite number of spatial-diffeomorphism invariant phase space functionals is extracted, and it is shown that these are observables in the sense that they Poisson commute with the reduced Hamiltonian. An infinite set of observables that have SL(2) indices are also found. This determination of the explicit time dependence of an infinite set of spatial-diffeomorphism invariant observables amounts to the solutions of the Hamiltonian Einstein equations for these observables.