The Einstein–Maxwell-particle system in the York canonical basis of ADM tetrad gravity. Part 2. The weak field approximation in the 3-orthogonal gauges and Hamiltonian post-minkowskian gravity: the N-body problem and gravitational waves with asymptotic background 1This paper is one of three companion papers published in the same issue of Can. J. Phys
Canadian journal of physics (Print)
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 3 K (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
... put equal to an arbitrary numerical function. The matter are point particles, with a Grassmann regularization of self-energies, and the electro-magnetic field in the radiation gauge: a ultraviolet cutoff 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) propagating on dynamically determined non-Euclidean 3-spaces. The conserved ADM energy and the Grassmann regularizzation of self-energies imply the correct energy balance. A generalized transverse-traceless gauge can be identified and the main tools for the detection of gravitational waves are reproduced 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 gauge problem in general relativity to understand which type of experimental observations may lead to a preferred choice for the inertial gauge variable 3 K in the PM space-times. In the third paper we will show that this choice is connected with the problem of dark matter.