Groupoid symmetry and constraints in general relativity

Understand

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.