The Einstein-Maxwell-Particle System: II) The Weak Field Approximation
  in the Non-Harmonic 3-Orthogonal Gauges and Hamiltonian Post-Minkowskian
  Gravity: the N-Body Problem and Gravitational Waves with Asymptotic
  Background
  In this second paper we define a Post-Minkowskian (PM) weak field
approximation leading to a linearization of the Hamilton equations of ADM
tetrad gravity in the York canonical basis in a family of non-harmonic
3-orthogonal Schwinger time gauges. The York time ${}^3K$ (the relativistic
inertial gauge variable, not existing in Newtonian gravity, parametrizing the
family and connected to the freedom in clock synchronization, i.e. to the
definition of the instantaneous 3-spaces) is put equal to an arbitrary
numerical function. A ultraviolet cutoff on the matter allows a consistent
linearization, which is shown to be the lowest order of a Hamiltonian
Post-Minkowskian (HPM) expansion. We solve the constraints and the Hamilton
equations for the tidal variables and we find Post-Minkowskian gravitational
waves with asymptotic background (and the correct quadrupole emission formula
and energy balance) propagating on dynamically determined non-Euclidean
3-spaces in these non-harmonic gauges. In conclusion we get a PM solution for
the gravitational field and we identify a class of PM Einstein space-times,
which will be studied in more detail in a third paper together with the PM
equations of motion for the particles and their Post-Newtonian expansion (but
in absence of the electro-magnetic field). Finally we make a discussion on the
{\it gauge problem in general relativity} to understand which type of
experimental observations may lead to a preferred choice for the inertial gauge
variable ${}^3K$ in the PM space-times. In the third paper we will show that
this choice is connected with the problem of dark matter.
