Embedding variables in finite dimensional models

TL;DR

This paper investigates the transformation between ADM and Kuchař variables in finite-dimensional models, demonstrating existence, inequivalence, and stability properties, with implications for understanding gravitational models and their variable embeddings.

Contribution

It provides a detailed analysis of the transformation properties between ADM and Kuchař variables in specific models, highlighting existence, inequivalence, and stability aspects.

Findings

Transformations to Kuchař variables exist on the entire ADM phase space for the Friedmann model.

The 2+1 gravity model admits an embedding variable description everywhere, even at symmetric points.

The new constraint surface is free from conical singularities and the equations are linearization stable.

Abstract

Global problems associated with the transformation from the Arnowitt, Deser and Misner (ADM) to the Kucha\v{r} 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 Kucha\v{r} 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 Kucha\v{r} 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 Kucha\v{r} 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 Kucha\v{r} description.