Kinematical Gravitational Charge Algebra

TL;DR

This paper demonstrates that the kinematical charges in the phase space of general relativity form a local Poincaré ISU(2) algebra, supporting the concept of Poincaré charge networks in discretized gravity.

Contribution

It proves that boundary charges associated with gauge and diffeomorphism constraints generate a local Poincaré algebra, linking boundary symmetries to discretized gravity frameworks.

Findings

Gauss and diffeomorphism constraints as conservation laws for boundary charges

Boundary charges generate a local Poincaré ISU(2) algebra

Supports Poincaré charge networks in discretized general relativity

Abstract

When formulated in terms of connection and coframes, and in the time gauge, the phase space of general relativity consists of a pair of conjugate fields: the flux 2-form and the Ashtekar connection. On this phase-space, one has to impose the Gauss constraints, the vector, and scalar Hamiltonian constraints. These are respectively generating local SU(2) gauge transformations, spatial diffeomorphisms, and time diffeomorphisms. We write the Gauss and space diffeomorphism constraints as conservation laws for a set of boundary charges, representing spin and momenta, respectively. We prove that these kinematical charges generate a local Poincar\'e ISU(2) symmetry algebra. This gives strong support to the recent proposal of Poincar\'e charge networks as a new realm for discretized general relativity [Classical Quantum Gravity 36, 195014 (2019)].