Hypersurface-deformation algebroids and effective spacetime models

Abstract

In canonical gravity, covariance is implemented by brackets of hypersurface-deformation generators forming a Lie algebroid. Lie algebroid morphisms therefore allow one to relate different versions of the brackets that correspond to the same space-time structure. An application to examples of modified brackets found mainly in models of loop quantum gravity can in some cases map the space-time structure back to the classical Riemannian form after a field redefinition. For one type of quantum corrections (holonomies), signature change appears to be a generic feature of effective space-time, and is shown here to be a new quantum space-time phenomenon which cannot be mapped to an equivalent classical structure. In low-curvature regimes, our constructions prove the existence of classical space-time structures assumed elsewhere in models of loop quantum cosmology, but also shows the existence of additional quantum corrections that have not always been included.