Canonical ADM Tetrad Gravity: from Metrological Inertial Gauge Variables to Dynamical Tidal Dirac observables

Dirac constraint theory allows to identify the York canonical basis (diagonalizing the York-Lichnerowicz approach) in ADM tetrad gravity for asymptotically Minkowskian space-times without super-translations. This allows to identify the inertial (gauge) and tidal (physical) degrees of freedom of the gravitational field and to interpret Ashtekar variables in these space-times. The use of radar 4-coordinates centered on a time-like observer allows to connect the 3+1 splittings of space-time with the relativistic metrology used in atomic physics and astronomy. The asymptotic ADM Poincar\'e group replaces the Poincar\'e group of particle physics. The general relativistic remnant of the gauge freedom in clock synchronization is described by the inertial gauge variable ${}^3K$, the trace of the extrinsic curvature of the non-Euclidean 3-spaces. The theory can be linearized in a Post-Minkowskian way by using the asymptotic Minkowski metric as an asymptotic background at spatial infinity and the family of non-harmonic 3-orthogonal Schwinger time gauges allows to reproduce the known results on gravitational waves in harmonic gauges. It is shown that the main signatures for the existence of dark matter can be reinterpreted as an relativistic inertial effect induced by ${}^3K$: while in the space-time inertial and gravitational masses coincide (equivalence principle), this is not true in the non-Euclidean 3-spaces (breaking of Newton equivalence principle), where the inertial mass has extra ${}^3K$-dependent terms simulating dark matter. Therefore a Post-Minkowskian extension of the existing Post-Newtonian celestial reference frame is needed.