Canonical Gravity in Degenerate Limit

TL;DR

This paper explores a degenerate limit of Hamiltonian gravity where the spatial metric determinant approaches zero, simplifying constraints and revealing connections to Carrollian gravity with a new geometric interpretation.

Contribution

It introduces two realizations of the degenerate limit in the Barbero-Immirzi SU(2) formulation, simplifying the Hamiltonian constraint and linking to Carrollian gravity.

Findings

Hamiltonian constraint becomes polynomial or free of ordering ambiguity.

Spatial diffeomorphisms become trivial in the limit.

Carrollian gravity emerges as a special case with a new geometric interpretation.

Abstract

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).