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

TL;DR

This paper develops a linearized Post-Minkowskian approximation in ADM tetrad gravity using non-harmonic gauges, deriving gravitational waves with an asymptotic background and exploring gauge choices related to dark matter.

Contribution

It introduces a Hamiltonian Post-Minkowskian framework in non-harmonic gauges, solving constraints and deriving gravitational waves with a focus on gauge variables and their physical implications.

Findings

Derived Post-Minkowskian gravitational waves with quadrupole emission.

Solved Hamilton equations for tidal variables in the weak field limit.

Identified gauge-dependent aspects potentially linked to dark matter.

Abstract

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.