Covariance and time regained in canonical general relativity

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 and ideas developed within history phase-space formalisms in [I. Kouletsis and K. V. Kucha\ifmmode \check{r}\else \v{r}\fi{}, Phys. Rev. D 65, 125026 (2002)], [K. Savvidou, Classical Quantum Gravity 18, 3611 (2001)], [K. Savvidou, Classical Quantum Gravity 21, 615 (2004)], [K. Savvidou, Classical Quantum Gravity 21, 631 (2004)], the time mapping $\mathbf{T}$: $\mathcal{M}\ensuremath{\rightarrow}\mathbb{R}$ and the space mapping $\mathbf{X}$: $\mathcal{M}\ensuremath{\rightarrow}\ensuremath{\Sigma}$ that define the Dirac--Arnowitt-Deser-Misner (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. The equal-time phase space $\mathcal{P}={{g}_{ij},{p}^{ij},{Y}^{\ensuremath{\alpha}},{P}_{\ensuremath{\alpha}}}$ includes the embeddings ${Y}^{\ensuremath{\alpha}}$: $\ensuremath{\Sigma}\ensuremath{\rightarrow}\mathcal{M}$ and their conjugate momenta ${P}_{\ensuremath{\alpha}}$. It is constrained by eight first-class constraints. The constraint surface $\mathcal{C}$ is determined by the super-Hamiltonian and supermomentum constraints of vacuum gravity and the vanishing of the embedding momenta. Deformations of the time and space mappings, $\ensuremath{\delta}\mathbf{T}$ and $\ensuremath{\delta}\mathbf{X}$, and spacetime diffeomorphisms, $V\ensuremath{\in}\mathrm{LDiff}\mathcal{M}$, induce symplectic diffeomorphisms of $\mathcal{P}$. While the generator ${\mathcal{D}}_{(\ensuremath{\delta}\mathbf{T},\ensuremath{\delta}\mathbf{X})}$ of deformations depends on all eight constraints, the generator ${\mathcal{D}}_{V}$ of spacetime diffeomorphisms depends only on the embedding momentum constraints. As a result, spacetime observables, namely, dynamical variables $F$ on $\mathcal{P}$ that are invariant under spacetime diffeomorphisms, ${F,{\mathcal{D}}_{V}}{\ensuremath{\mid}}_{\mathcal{C}}=0$, are not necessarily invariant under the deformations of the mappings, ${F,{\mathcal{D}}_{(\ensuremath{\delta}\mathbf{T},\ensuremath{\delta}\mathbf{X})}}{\ensuremath{\mid}}_{\mathcal{C}}\ensuremath{\ne}0$, nor are they constants of the motion, ${F,\ensuremath{\int}{d}^{3}x\mathcal{H}}{\ensuremath{\mid}}_{\mathcal{C}}\ensuremath{\ne}0$. Dirac observables form only a subset of spacetime observables that are invariant under the transformations of $\mathbf{T}$ and $\mathbf{X}$ and do not evolve in time. In this generally covariant framework, the conventional interpretation of the canonical theory, due to Bergmann and Dirac, amounts to postulating that the transformations of the reference system $(\mathbf{T},\mathbf{X})$ have no measurable consequences; i.e., that all first-class constraints generate gauge transformations. If this postulate is not deemed necessary, canonical gravity admits no classical problem of time.