Covariance and Time Regained in Canonical General Relativity

Abstract

Canonical vacuum gravity is expressed in generally-covariant form in order that spacetime diffeomorphisms be represented within its equal-time phase space. In accordance with the principle of general covariance, the time mapping ${\T}: {\yman} \to {\rman}$ and the space mapping ${\X}: {\yman} \to {\xman}$ that define the Dirac-ADM foliation are incorporated into the framework of the Hilbert variational principle. The resulting canonical action encompasses all individual Dirac-ADM actions, corresponding to different choices of foliating vacuum spacetimes by spacelike hypersurfaces. In this framework, spacetime observables, namely, dynamical variables that are invariant under spacetime diffeomorphisms, are not necessarily invariant under the deformations of the mappings $\T$ and $\X$, nor are they constants of the motion. Dirac observables form only a subset of spacetime observables that are invariant under the transformations of $\T$ and $\X$ and do not evolve in time. The conventional interpretation of the canonical theory, due to Bergmann and Dirac, can be recovered only by postulating that the transformations of the reference system $({\T},{\X})$ have no measurable consequences. If this postulate is not deemed necessary, covariant canonical gravity admits no classical problem of time.