A discrete discontinuity between the two phases of gravity

Abstract

When tetrad (metric) fields are not invertible, the standard canonical formulation of gravity cannot be adopted as it is. Here we develop a Hamiltonian theory of gravity for non-invertible tetrad. In contrast to Einstein gravity, this phase is found to exhibit three local degrees of freedom. This reflects a discrete discontinuity in the limit of a vanishing tetrad determinant. For the particular case of vanishing lapse, the Hamiltonian constraint disappears from the classical theory upon fixing the torsional gauge-freedom. Any state functional invariant under the internal gauge rotations and spatial diffeomorphisms is a formal solution of the associated quantum theory. The formulation here provides a Hamiltonian basis to analyze gravity theory around a physical singularity, which corresponds to a zero of the tetrad determinant in curved spacetime.