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

TL;DR

This paper develops a background-independent Hamiltonian linearization of tetrad gravity in a fixed gauge, describing gravitational waves through Dirac observables without a background metric, and confirms they satisfy linearized Einstein equations.

Contribution

It introduces a novel Hamiltonian linearization method in a fixed gauge, avoiding background metrics, and explicitly solves all constraints to describe gravitational waves as Dirac observables.

Findings

Derivation of wave equations for Dirac observables

Validation that linearized Einstein equations are satisfied

Analysis of geodesic and geodesic deviation equations

Abstract

In the framework of the rest-frame instant form of tetrad gravity, where the Hamiltonian is the weak ADM energy ${\hat E}_{ADM}$, we define a special completely fixed 3-orthogonal Hamiltonian gauge, corresponding to a choice of {\it 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) $r_{\bar a}(\tau ,\vec \sigma)$, $\pi_{\bar a}(\tau ,\vec \sigma)$, $\bar a = 1,2$. We define a Hamiltonian linearization of the theory, i.e. gravitational waves, {\it 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 ${\hat E}_{ADM}$. {\it 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 $r_{\bar a}(\tau ,\vec \sigma)$, which replace the two polarizations of the TT harmonic gauge, and that {\it linearized Einstein's equations are satisfied} . Finally we study the geodesic equation, both for time-like and null geodesics, and the geodesic deviation equation.