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Authors: David Alba, Luca Lusanna

Date: 26 Mar 2010

Abstract: In this second paper we define a Post-Minkowskian 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 ${}ˆ3K$ (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. 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. Then a Post-Newtonian (PN) expansion at all orders of HPM can be done by adding suitable slow motion conditions.
The dependence on the York time of the equations of motion of the particles and of quantities like the redshift and the luminosity distance is explicitly given. As a consequence of a discussion on the {\it gauge problem in general relativity}, it turns out that there is the possibility that at least part of dark matter could be explained as a relativistic inertial effect at the 0.5PN order induced by the York time.

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