Internal Time Formalism for Spacetimes with Two Killing Vectors

TL;DR

This paper develops an internal time formalism for spacetimes with two Killing vectors, extending the phase space and identifying gauge-invariant canonical variables to address the problem of time in canonical gravity.

Contribution

It introduces a method to define internal spacetime coordinates in symmetric spacetimes, extending the phase space and establishing gauge-invariant canonical transformations.

Findings

Canonical variables for internal time are identified.

Models are mapped to harmonic map field theories.

Problems of time are partially resolved in these models.

Abstract

The Hamiltonian structure of spacetimes with two commuting Killing vector fields is analyzed for the purpose of addressing the various problems of time that arise in canonical gravity. Two specific models are considered: (i) cylindrically symmetric spacetimes, and (ii) toroidally symmetric spacetimes, which respectively involve open and closed universe boundary conditions. For each model canonical variables which can be used to identify points of space and instants of time, {\it i.e.}, internally defined spacetime coordinates, are identified. To do this it is necessary to extend the usual ADM phase space by a finite number of degrees of freedom. Canonical transformations are exhibited that identify each of these models with harmonic maps in the parametrized field theory formalism. The identifications made between the gravitational models and harmonic map field theories are completely gauge invariant, that is, no coordinate conditions are needed. The degree to which the problems of time are resolved in these models is discussed.