EINSTEINS EQUATIONS AND THE CHIRAL MODEL

Authors
Citation
V. Husain, EINSTEINS EQUATIONS AND THE CHIRAL MODEL, Physical review. D. Particles and fields, 53(8), 1996, pp. 4327-4334
Citations number
35
Categorie Soggetti
Physics, Particles & Fields
ISSN journal
05562821
Volume
53
Issue
8
Year of publication
1996
Pages
4327 - 4334
Database
ISI
SICI code
0556-2821(1996)53:8<4327:EEATCM>2.0.ZU;2-Y
Abstract
The vacuum Einstein equations for spacetimes with two commuting spacel ike Killing field symmetries are studied using the Ashtekar variables. The case of compact spacelike hypersurfaces which are three-tori is c onsidered, and the determinant of the Killing two-torus metric is chos en as the time gauge. The Hamiltonian evolution equations in this gaug e may be rewritten as those of a modified SL(2) principal chiral model with a time-dependent ''coupling constant,'' or equivalently, with ti me-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-invar iant phase-space functionals is extracted, and it is shown that these are observables in the sense that they Poisson commute with the reduce d Hamiltonian. An infinite set of observables that have SL(2) indices is also found. This determination of the explicit time dependence of a n infinite set of spatial-diffeomorphism-invariant observables amounts to the solutions of the Hamiltonian Einstein equations for these obse rvables.