Groupoid symmetry and constraints in general relativity. 1: kinematics

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