The Einstein-Maxwell-Particle System in the York Canonical Basis of ADM
  Tetrad Gravity: I) The Equations of Motion in Arbitrary Schwinger Time Gauges
  We study the coupling of N charged scalar particles plus the electro-magnetic
field to ADM tetrad gravity and its canonical formulation in asymptotically
Minkowskian space-times without super-translations. We make the canonical
transformation to the York canonical basis, where there is a separation between
the {\it inertial} (gauge) variables and the {\it tidal} ones inside the
gravitational field and a special role of the Eulerian observers associated to
the 3+1 splitting of space-time. The Dirac Hamiltonian is weakly equal to the
weak ADM energy. The Hamilton equations in Schwinger time gauges are given
explicitly. In the York basis they are naturally divided in four sets: a) the
contracted Bianchi identities; b) the equations for the inertial gauge
variables; c) the equations for the tidal ones; d) the equations for matter.
Finally we give the restriction of the Hamilton equations and of the
constraints to the family of {\it non-harmonic 3-orthogonal} gauges, in which
the instantaneous Riemannian 3-spaces have a diagonal 3-metric. The non-fixed
inertial gauge variable ${}^3K$ (the freedom in the clock synchronization
convention) gives rise to a negative kinetic term in the weak ADM energy
vanishing only in the gauges with ${}^3K = 0$: is it relevant for dark energy
and back-reaction? In the second paper there will be the linearization of the
theory to obtain Hamiltonian post-Minkowskian gravity with asymptotic Minkowski
background, non-flat instantaneous 3-spaces and no post-Newtonian expansion.
This will allow to explore the inertial effects induced by the York time
${}^3K$ in non-flat 3-spaces and to check how much dark matter can be explained
as an inertial aspect of Einstein's general relativity.
