Internal Time Formalism for Spacetimes with Two Killing Vectors

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.

36 pages, TeX

References in corpus (4)

• Geometrodynamics of Schwarzschild Black Holes
• On "asymptotically flat" space-times with $G_2$-invariant Cauchy surfaces
• A Striking Property of the Gravitational Hamiltonian
• Is General Relativity an 'Already Parametrized' Field Theory?

Cited by in corpus (25)

• The Asymptotic Safety Scenario in Quantum Gravity -- An Introduction
• Probing Quantum Gravity Through Exactly Soluble Midi-Superspaces I
• The Principle of Symmetric Criticality in General Relativity
• Selected solutions of Einstein's field equations: their role in general relativity and astrophysics
• Renormalization and asymptotic safety in truncated quantum Einstein gravity
• Quantization of Midisuperspace Models
• Quantum Fields at Any Time
• Quantum Dynamics of the Polarized Gowdy Model
• Midi-superspace Models of Canonical Quantum Gravity
• Hamiltonian thermodynamics of a Lovelock black hole
• Microcausality and Quantum Cylindrical Gravitational Waves
• Hamiltonian Dynamics of Linearly Polarized Gowdy Models Coupled to Massless Scalar Fields
• Functional evolution of quantum cylindrical waves
• Gauge fixing and the Hamiltonian for cylindrical spacetimes
• Observables for the polarized Gowdy model
• Asymptotic behaviour of cylindrical waves interacting with spinning strings
• Asymptotic Analysis of Field Commutators for Einstein-Rosen Gravitational Waves
• Observable effect of quantized cylindrical gravitational waves
• Particles and vacuum for perturbative and non-perturbative Einstein-Rosen gravity
• Asymptotics of Regulated Field Commutators for Einstein-Rosen Waves
• Quantum Einstein-Rosen waves: Coherent states and n-point functions
• Emergence of General Relativity from Loop Quantum Gravity: A Summary
• Quantum Cauchy Surfaces in Canonical Quantum Gravity
• Quantum Reference Frames via Transition Amplitudes in Timeless Quantum Gravity
• Harvesting entanglement from the cylindrical gravitational wave spacetime