Narrow Results

The theory of cosmological perturbations is a well-elaborated field and has been successfully applied, e.g. to model the structure formation in our universe and the prediction of the power spectrum of the cosmic microwave background. To deal with the diffeomorphism invariance of general relativity, one generally introduces combinations of the metric and matter perturbations, which are gauge invariant up to the considered order in the perturbations. For linear cosmological perturbations, one works with the so-called Bardeen potentials widely used in this context. However, there exists no common procedure to construct gauge invariant quantities also for higher-order perturbations. Usually, one has to find new gauge invariant quantities independently for each order in perturbation theory. With the relational formalism introduced by Rovelli and further developed by Dittrich and Thiemann, it is in principle possible to calculate manifestly gauge invariant quantities, that is quantities that are gauge invariant up to arbitrary order once one has chosen a set of so-called reference fields, often also called clock fields. This article contains a review of the relational formalism and its application to canonical general relativity following the work of Garcia, Pons, Sundermeyer and Salisbury. As the starting point for our application of this formalism to cosmological perturbation theory, we also review the Hamiltonian formulation of the linearized theory for perturbations around FLRW spacetimes. The main aim of our work will be to identify clock fields in the context of the relational formalism that can be used to reconstruct quantities like the Bardeen potential as well as the Mukhanov–Sasaki variable. This requires a careful analysis of the canonical formulation in the extended ADM-phase-space where lapse and shift are treated as dynamical variables. The actual construction of such observables and further investigations thereof will be carried out in our companion paper.

We find the Hamiltonian expression in the York basis of canonical ADM tetrad gravity of the 4-Weyl tensor of the asymptotically Minkowskian space-time. Like for the 4-Riemann tensor we find a radar tensor (whose components are 4-scalars due to the use of radar 4-coordinates), which coincides with the 4-Weyl tensor on-shell on the solutions of Einstein's equations. Then, by using the Hamiltonian null tetrads, we find the Hamiltonian expression of the Weyl scalars of the Newman–Penrose approach and of the four eigenvalues of the 4-Weyl tensor. After having introduced the Dirac observables (DOs) of canonical gravity, whose determination requires the solution of the super-Hamiltonian and super-momentum constraints, we discuss the connection of the DOs with the notion of 4-scalar Bergmann observables (BOs). Due to the use of radar 4-coordinates these two types of observables coincide in our formulation of canonical ADM tetrad gravity. However, contrary to Bergmann proposal, the Weyl eigenvalues are shown not to be BOs, so that their relevance is only in their use (first suggested by Bergmann and Komar) for giving a physical identification as point-events of the mathematical points of the space-time 4-manifold. Finally we give the expression of the Weyl scalars in the Hamiltonian post-Minkowskian linearization of canonical ADM tetrad gravity in the family of (non-harmonic) 3-orthogonal Schwinger time gauges.

In this updated review of canonical ADM tetrad gravity in a family of globally hyperbolic asymptotically Minkowskian space-times without super-translations I show which is the status-of-the-art in the search of a canonical basis adapted to the first-class Dirac constraints and of the Dirac observables of general relativity (GR) describing the tidal degrees of freedom of the gravitational field. In these space-times the asymptotic ADM Poincaré group replaces the Poincaré group of particle physics, there is a York canonical basis diagonalizing the York–Lichnerowicz approach and a post-Minkowskian linearization is possible with the associated description of gravitational waves in the family of non-harmonic 3-orthogonal Schwinger time gauges. Moreover I show that every fixation of the inertial gauge variables (i.e. the choice of a non-inertial frame) of every generally covariant formulation of GR is equivalent to a set of conventions for the metrology of the space-time (like the GPS ones near the Earth): for instance the freedom in clock synchronization is described by the inertial gauge variable York time (the trace of the extrinsic curvature of the instantaneous 3-spaces). This inertial gauge freedom and the non-Euclidean nature of the instantaneous 3-spaces required by the equivalence principle are connected with the dark side of the universe and could explain the presence of dark matter or at least part of it by means of the adoption of suitable metrical conventions for the ICRS celestial reference system. Also some comments on a canonical quantization of GR coherent with this viewpoint are done.