The post-Minkowskian limit of ADM tetrad gravity in the 3-orthogonal gauges of the non-inertial rest-frames is defined with particles and the electromagnetic field as matter. Then, the post-Newtonian expansion of the post-Minkowskian linearization is studied. For binaries, the results are compatible with the standard one in harmonic gauges. However, there is the new result that a non-local version of the inertial gauge variable York time may explain many of the experimental data giving rise to the existence of dark matter, which would be reduced to a relativistic inertial effect to be treated by means of relativistic celestial metrology.

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.