Groupoid symmetry and constraints in general relativity

TL;DR

This paper explores the algebraic structure of the constraints in general relativity, revealing their relation to the Lie algebroid of a groupoid of diffeomorphisms between hypersurfaces, independent of Einstein equations.

Contribution

It demonstrates that the constraint brackets in general relativity match those in a Lie algebroid of a diffeomorphism groupoid, offering a new geometric perspective.

Findings

Constraint brackets match Lie algebroid relations.

Constraints relate to a groupoid of hypersurface diffeomorphisms.

Direct link between constraints and Einstein equations remains open.

Abstract

When the vacuum Einstein equations are cast in the form of hamiltonian evolution equations, the initial data lie in the cotangent bundle of the manifold M\Sigma\ of riemannian metrics on a Cauchy hypersurface \Sigma. As in every lagrangian field theory with symmetries, the initial data must satisfy constraints. But, unlike those of gauge theories, the constraints of general relativity do not arise as momenta of any hamiltonian group action. In this paper, we show that the bracket relations among the constraints of general relativity are identical to the bracket relations in the Lie algebroid of a groupoid consisting of diffeomorphisms between space-like hypersurfaces in spacetimes. A direct connection is still missing between the constraints themselves, whose definition is closely related to the Einstein equations, and our groupoid, in which the Einstein equations play no role at all. We discuss some of the difficulties involved in making such a connection.