Gravitational Constraint Combinations Generate a Lie Algebra

TL;DR

This paper derives a differential equation for scalar combinations of gravitational constraints that form a Lie algebra, potentially simplifying the algebraic structure of canonical gravity.

Contribution

It introduces a new class of scalar constraints with Abelian Poisson brackets, generalizing Kuchař's approach and replacing the Dirac algebra with a true Lie algebra in canonical gravity.

Findings

New scalar combinations satisfy a specific PDE.

These combinations form an Abelian algebra with Poisson brackets.

The resulting algebra simplifies the structure of constraints in gravity.

Abstract

We find a first--order partial differential equation whose solutions are all ultralocal scalar combinations of gravitational constraints with Abelian Poisson brackets between themselves. This is a generalisation of the Kucha\v{r} idea of finding alternative constraints for canonical gravity. The new scalars may be used in place of the hamiltonian constraint of general relativity and, together with the usual momentum constraints, replace the Dirac algebra for pure gravity with a true Lie algebra: the semidirect product of the Abelian algebra of the new constraint combinations with the algebra of spatial diffeomorphisms.