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We study the coupling of N charged scalar particles plus the electro-magneticfield to ADM tetrad gravity and its canonical formulation in asymptoticallyMinkowskian space-times without super-translations. We make the canonicaltransformation to the York canonical basis, where there is a separation betweenthe {\it inertial} (gauge) variables and the {\it tidal} ones inside thegravitational field and a special role of the Eulerian observers associated tothe 3+1 splitting of space-time. The Dirac Hamiltonian is weakly equal to theweak ADM energy. The Hamilton equations in Schwinger time gauges are givenexplicitly. In the York basis they are naturally divided in four sets: a) thecontracted Bianchi identities; b) the equations for the inertial gaugevariables; c) the equations for the tidal ones; d) the equations for matter.Finally we give the restriction of the Hamilton equations and of theconstraints to the family of {\it non-harmonic 3-orthogonal} gauges, in whichthe instantaneous Riemannian 3-spaces have a diagonal 3-metric. The non-fixedinertial gauge variable ${}^3K$ (the freedom in the clock synchronizationconvention) gives rise to a negative kinetic term in the weak ADM energyvanishing only in the gauges with ${}^3K = 0$: is it relevant for dark energyand back-reaction? In the second paper there will be the linearization of thetheory to obtain Hamiltonian post-Minkowskian gravity with asymptotic Minkowskibackground, 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 explainedas an inertial aspect of Einstein's general relativity.

The Einstein-Maxwell-Particle System in the York Canonical Basis of ADM Tetrad Gravity: I) The Equations of Motion in Arbitrary Schwinger Time Gauges