In the chosen family of Einstein space-times one can give an ADM formulation of tetrad gravity and of its first-class constraints, so that fermions can be described in this framework. There are 16 configuration variables, 16 momenta, and 14 first-class constraints. Then one can define a new canonical basis adapted to 10 of the 14 constraints (not to the super-Hamiltonian and super-momentum ones) with a Shanmugadhasan canonical transformation. This allows identifying two pairs of canonical variables describing the tidal effects (the gravitational waves after linearization). However, they are not Dirac observables.

After a description of inertial and non-inertial frames in the Galilei space-time of non-relativistic Newtonian physics with a discussion of inertial forces, there is metrological definition of what is time and space in special relativity. Then there is a review of the standard 1+3 approach for the “local” description of non-inertial frames and of its limitations.

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.

Given an isolated system of either free or interacting particles and the associated realization of the ten conserved Poincaré generators its total conserved time-like 4-momentum defines its inertial rest-frame as the 3+1 splitting whose space-like 3-spaces (named Wigner 3-spaces) are orthogonal to it and whose inertial observer is the Fokker–Pryce 4-center of inertia. There is a discussion of the problem of the relativistic center of mass based on the fact that the 4-center functions “only” of the Poincaré generators of the isolated system are the following three non-local quantities: the non-canonical covariant Fokker–Pryce 4-center of inertia, the canonical non-covariant Newton–Wigner 4-center of mass and the non-canonical non-covariant Mőller 4-center of energy. At the Hamiltonian level one is able to express the canonical world-lines of the particles and their momenta in terms of the Jacobi variables of the external Newton–Wigner center of mass (a non-local non-covariant non-measurable quantity) and of Wigner-covariant relative 3-coordinates and 3-momenta inside the Wigner 3-spaces. This solves the problem of the elimination of relative times in relativistic bound states and to formulate a consistent Wigner-covariant relativistic quantum mechanics of point particles. The non-relativistic limit gives the Hamilton–Jacobi description of the system after the separation of Newtonian center of mass. Finally there is the definition of the non-inertial rest-frames whose 3-spaces are orthogonal to the total 4-momentum of the isolated system at spatial infinity.