The Geroch group in the Ashtekar formulation

Abstract

We study the Geroch group in the framework of the Ashtekar formulation. In the case of the one-Killing-vector reduction, it turns out that the third column of the Ashtekar connection is essentially the gradient of the Ernst potential, which implies that the both quantities are based on the ``same'' complexification. In the two-Killing-vector reduction, we demonstrate Ehlers' and Matzner-Misner's SL(2,R) symmetries, respectively, by constructing two sets of canonical variables that realize either of the symmetries canonically, in terms of the Ashtekar variables. The conserved charges associated with these symmetries are explicitly obtained. We show that the gl(2,R) loop algebra constructed previously in the loop representation is not the Lie algebra of the Geroch group itself. We also point out that the recent argument on the equivalence to a chiral model is based on a gauge-choice which cannot be achieved generically.

40 pages, revtex

References in corpus (4)

• Minisuperspaces: Observables and Quantization
• Observables for spacetimes with two Killing field symmetries
• Non-Perturbative Canonical Quantization of Minisuperspace Models: Bianchi Types I and II
• Bibliography of Publications related to Classical and Quantum Gravity in terms of the Ashtekar Variables

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