Action functionals of single scalar fields and arbitrary--weight gravitational constraints that generate a genuine Lie algebra

TL;DR

This paper explores how certain combinations of gravitational constraints, involving scalar densities of arbitrary weight, can be derived from an action principle with a scalar field, leading to a genuine Lie algebra structure.

Contribution

It demonstrates that alternative gravitational constraints forming a true Lie algebra can be obtained from a scalar field action with specific reality conditions.

Findings

Constraints form a true Lie algebra with Poisson brackets strongly vanishing.

Such constraints can be derived from a scalar field action with a non-derivative coupling.

The approach generalizes the algebraic structure of vacuum gravity constraints.

Abstract

We discuss the issue initiated by Kucha\v{r} {\it et al}, of replacing the usual Hamiltonian constraint by alternative combinations of the gravitational constraints (scalar densities of arbitrary weight), whose Poisson brackets strongly vanish and cast the standard constraint-system for vacuum gravity into a form that generates a true Lie algebra. It is shown that any such combination---that satisfies certain reality conditions---may be derived from an action principle involving a single scalar field and a single Lagrange multiplier with a non--derivative coupling to gravity.