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.
