The following family of Einstein space-times allows the use of the 3+1 approach: (1) globally hyperbolic (this allows the ADM Hamiltonian formulation); (2) asymptotically Minkowskian at spatial infinity (all the 3-spaces approach parallel space-like hyper-planes); (3) without super-translations (at spatial infinity there is the asymptotic ADM Poincaré algebra needed for particle physics). It turns out that the asymptotic ADM Poincaré 4-momentum is orthogonal to the asymptotic hyper-planes. Therefore, the 3+1 approach allows describing the Hamiltonian formulation of metric gravity and of its first-class constraints in the family of the non-inertial rest-frames.