Hamiltonian expression of curvature tensors in the York canonical basis: (II) Weyl tensor, Weyl scalars, Weyl eigenvalues and the problem of the observables of the gravitational field

Abstract

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 NewmanPerose approach and of the four eigenvalues of the 4-Weyl tensor.After having introduced the Dirac observables of canonical gravity, whose determination requires the solution of the super-Hamiltonian and super-momentum constraints, we discuss the connection of the Dirac observables with the notion of 4-scalar Bergmann observables. 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 Bergmann observables, 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.

From Relativistic Mechanics towards Relativistic Statistical Mechanics

“…In the family of globally-hyperbolic, asymptotically-Minkowskian space-times without super-translations discussed in [76][77][78][79][80][81][82][83], there is the possibility of defining the asymptotic ADM Poincaré generators at spatial infinity, where the instantaneous three-spaces Σ τ become Euclidean, for every kind of matter. The absence of super-translations implies that these three-spaces are relativistic non-inertial rest frames.…”

“…The four-coordinates σ A = (τ; σ r ) are named radar four-coordinates: they were introduced by Bondi [75] in general relativity and have been used for the definition of global non-inertial frames by means of 3 + 1 splittings of a certain family of Einstein space-times (see [76][77][78][79][80][81] and the reviews [82,83]). …”

Killing symmetries as Hamiltonian constraints

“…The Hamiltonian expression [20,21] of the extrinsic curvature 3 K rs of the instantaneous 3-spaces Σ τ and of the 4-Christoffel symbols 4…”

“…It is known from the work in Refs. [1][2][3][4][5] that for Einstein and Eistein-Maxwell theories gravity [15][16][17][18][19][20][21] (see Refs. [22] for reviews).…”

“…3 Analogously Moncrief [6] has shown that gauge symmetries and Gribov ambiguity are absent in certain weighted Sobolev spaces. gravity [15][16][17][18][19][20][21] (see Refs. [22] for reviews).…”