Covariance and Time Regained in Canonical General Relativity

TL;DR

This paper reformulates canonical vacuum gravity to include spacetime diffeomorphisms within the phase space, challenging traditional notions of observables and the problem of time in general relativity.

Contribution

It introduces a covariant canonical framework that incorporates spacetime mappings, expanding the understanding of observables and the classical problem of time.

Findings

Spacetime observables are not necessarily invariant under foliation deformations.

Dirac observables are a subset of spacetime observables invariant under certain transformations.

The classical problem of time can be avoided without additional postulates.

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.