Embedding variables in finite dimensional models

Abstract

Global problems associated with the transformation from the Arnowitt, Deser and Misner (ADM) to the Kuchař variables are studied. Two models are considered: The Friedmann cosmology with scalar matter and the torus sector of the 2+1 gravity. For the Friedmann model, the transformations to the Kuchař description corresponding to three different popular time coordinates are shown to exist on the whole ADM phase space, which becomes a proper subset of the Kuchař phase spaces. The 2+1 gravity model is shown to admit a description by embedding variables everywhere, even at the points with additional symmetry. The transformation from the Kuchař to the ADM description is, however, many-to-one there, and so the two descriptions are inequivalent for this model, too. The most interesting result is that the new constraint surface is free from the conical singularity and the new dynamical equations are linearization stable. However, some residual pathology persists in the Kuchař description.