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When the vacuum Einstein equations are cast in the form of hamiltonianevolution equations, the initial data lie in the cotangent bundle of themanifold M\Sigma\ of riemannian metrics on a Cauchy hypersurface \Sigma. As inevery lagrangian field theory with symmetries, the initial data must satisfyconstraints. But, unlike those of gauge theories, the constraints of generalrelativity do not arise as momenta of any hamiltonian group action. In thispaper, we show that the bracket relations among the constraints of generalrelativity are identical to the bracket relations in the Lie algebroid of agroupoid consisting of diffeomorphisms between space-like hypersurfaces inspacetimes. A direct connection is still missing between the constraintsthemselves, whose definition is closely related to the Einstein equations, andour groupoid, in which the Einstein equations play no role at all. We discusssome of the difficulties involved in making such a connection.

Groupoid symmetry and constraints in general relativity