Post-Minkowskian Gravity: Dark Matter as a Relativistic Inertial Effect?
  A review is given of the theory of non-inertial frames (with the associated
inertial effects and the study of the non-relativistic limit) in Minkowski
space-time, of parametrized Minkowski theories and of the rest-frame instant
form of dynamics for isolated systems admitting a Lagrangian description. The
relevance and gauge equivalence of the clock synchronization conventions for
the identification of the instantaneous 3-spaces (Euclidean only in inertial
frames) are described. Then this formalism is applied to tetrad gravity in
globally hyperbolic, asymptotically Minkowskian space-times without
super-translations, where the equivalence principle implies the absence of
global inertial frames. The recently discovered York canonical basis,
diagonalizing the York-Lichnerowicz approach, allows to identify the gauge
variables (inertial effects in general relativity) and the tidal ones (the
gravitational waves of the linearized theory) and to clarify the meaning of the
Hamilton equations. The role of the gauge variable ${}^3K$, the trace of the
extrinsic curvature of the non-Euclidean 3-space (the York time not existing in
Newton theory), as a source of inertial effects is emphasized. After the
presentation of preliminary results on the linearization of tetrad gravity in
the family of non-harmonic 3-orthogonal gauges with a free value of ${}^3K$, we
define post-Minkowskian gravitational waves (without post-Newtonian
approximations on the matter sources) propagating in a non-Euclidean 3-space,
emphasizing the non-graviton-like aspects of gravity. It is conjectured that
dark matter may be explained as a relativistic inertial effect induced by
${}^3K$: it would simulate the need to choose a privileged gauge connected with
the observational conventions for the description of matter.
