Hamiltonian linearization of the rest-frame instant form of tetrad gravity in a completely fixed 3-orthogonal gauge: A radiation gauge for background-independent gravitational waves in a post-Minkowskian Einstein spacetime
General Relativity and Gravitation vol:36 issue:5 pages:1055-1134
In the framework of the rest-frame instant form of tetrad gravity, where the Hamiltonian is the weak ADM energy (E) over cap (ADM), we define a special completely fixed 3-orthogonal Hamiltonian gauge, corresponding to a choice of non-harmonic 4-coordinates, in which the independent degrees of freedom of the gravitational field are described by two pairs of canonically conjugate Dirac observables (DO) r((a) over bar)(tau, (σ) over right arrow), pi((a) over bar)(tau, (σ) over right arrow), (a) over bar = 1, 2. We define a Hamiltonian linearization of the theory, i. e. gravitational waves, without introducing any background 4-metric, by retaining only the linear terms in the DO's in the super-hamiltonian constraint (the Lichnerowicz equation for the conformal factor of the 3-metric) and the quadratic terms in the DO's in (E) over cap (ADM) . We solve all the constraints of the linearized theory: this amounts to work in a well defined post-Minkowskian Christodoulou-Klainermann space-time. The Hamilton equations imply the wave equation for the DO's r((a) over bar)(tau, (σ) over right arrow), which replace the two polarizations of theTTharmonic gauge, and that linearized Einstein's equations are satisfied. Finally we study the geodesic equation, both for time-like and null geodesics, and the geodesic deviation equation.