Canonical gravity in degenerate limit

We construct a limit of Hamiltonian gravity as the determinant of the spatial triad (and hence of the four-metric) goes to zero. Within the Barbero–Immirzi SU(2) formulation, we present two possible realizations of this limit, with the consequence that the Hamiltonian constraint becomes simpler and spatial diffeomorphisms become trivial. In the first case, the Hamiltonian constraint exhibits a polynomial structure, being formally similar to the Euclidean Hamitonian constraint of Sen–Ashtekar self-dual formulation. In the latter, the constraints become free from ordering ambiguity. Further, we show that the Carrollian gravity emerges as a special case of this degenerate limit, thus providing it a new geometric interpretation independent of the speed of light or any dimensionful coupling constant (G).