Gravitational Constraints which Generate a Lie Algebra

TL;DR

This paper introduces new gravitational constraints that form a true Lie algebra, derived from coupling gravity to dust and scalar fields, simplifying the algebraic structure of the theory.

Contribution

It presents novel quadratic combinations of gravitational constraints that generate a Lie algebra, improving the understanding of gravitational symmetries.

Findings

Quadratic combinations of super-Hamiltonian and supermomentum vanish in Poisson brackets.

Coupling to dust yields simple quadratic constraints with Lie algebra structure.

Coupling to scalar fields produces similar constraints, though less simple.

Abstract

The coupling of gravity to dust helps to discover simple quadratic combinations of the gravitational super-Hamiltonian and supermomentum whose Poisson brackets strongly vanish. This leads to a new form of vacuum constraints which generate a true Lie algebra. We show that the coupling of gravity to a massless scalar field leads to yet another set of constraints with the same property, albeit not as simple as that based on the coupling to dust.