Internal time formalism for spacetimes with two Killing vectors

Joseph D. Romano, Charles G. Torre

Research output: Contribution to journalArticlepeer-review

39 Scopus citations


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, 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.

Original languageEnglish
Pages (from-to)5634-5650
Number of pages17
JournalPhysical Review D - Particles, Fields, Gravitation and Cosmology
Issue number10
StatePublished - 1996


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