Abstract
In canonical gravity, general covariance is implemented by hypersurface-deformation symmetries on thephase space. The different versions of hypersurface deformations required for full covariance have complicated interplays with one another, governed by non-Abelian brackets with structure functions. For spherically symmetric space-times, it is possible to identify a certain Abelian substructure within general hypersurface deformations, which suggests a simplified realization as a Lie algebra. The generators of this substructure can be quantized more easily than full hypersurface deformations, but the symmetries they generate do not directly correspond to hypersurface deformations. The availability of consistent quantizations therefore does not guarantee general covariance or a meaningful quantum notion thereof. In addition to placing the Abelian substructure within the full context of spherically symmetric hypersurface deformation, this paper points out several subtleties relevant for attempted applications in quantized space-time structures. In particular, it follows that recent constructions by Gambini, Olmedo, and Pullin in an Abelianized setting fail to address the covariance crisis of loop quantum gravity.
Highlights
Canonical gravity describes the 4-dimensional, generally covariant structure of spacetime by canonical fields defined on the slices of a spatial foliation
The free constant in (19) can be determined through boundary conditions, which would restrict the lapse functions allowed in gauge transformations. It seems that the partial Abelianization eliminates structure functions from the brackets and may simplify quantization and the preservation of symmetries and covariance
Restricting the system to phase-space independent L, by contrast, implies that the transformation from the original hypersurface-deformation structure to the brackets of D [ M ] and C [ L] is not invertible. It is unclear whether hypersurface deformations and general covariance can be recovered from a partial Abelianization, in particular if the latter has been modified by quantum corrections
Summary
Canonical gravity describes the 4-dimensional, generally covariant structure of spacetime by canonical fields defined on the slices of a spatial foliation The evolution of these fields in time as well as transformations between different foliations are described by the geometrical structure of hypersurface deformations. The reference to normal and tangential directions relative to a foliation implies crucial differences between the mathematical formulation of hypersurface deformations in canonical gravity and the more common formulation of general covariance in terms of space-time tensors. The full structure of transformations is required for general covariance to be implemented properly in the solutions of a canonical theory of gravity, in particular one that has been quantized, modified or deformed by new physical effects. Using our results about general hypersurface deformation structures, we will explain why the covariance claims of [8] cannot hold
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