Dynamical time and Ashtekar variables for the Husain-Kucha\v{r} model

TL;DR

This paper explores the connection between the Husain-Kuchař model, dynamical time variables, and Ashtekar variables using a geometric Hamiltonian approach, offering insights into the dynamics of general relativity.

Contribution

It introduces a geometric Hamiltonian framework for the Husain-Kuchař model with dynamical time and relates it to Ashtekar variables without fixing a time gauge.

Findings

Clarifies the role of time in the Husain-Kuchař model

Provides a geometric Hamiltonian formulation for the model

Links Ashtekar variables to dynamical time approaches

Abstract

The relation between the Husain-Kucha\v{r} model and some extensions thereof that incorporate a dynamical time variable is explored. To this end, we rely on the geometric approach to the Hamiltonian dynamics of singular (constrained) systems provided by the Gotay-Nester-Hinds method (GNH). We also use recent results regarding the derivation of the Ashtekar formulation for Euclidean general relativity without using the time gauge. The insights provided by the geometric approach followed in the paper shed light on several issues related to the dynamics of general relativity and the role of time.