The physical role of gravitational and gauge degrees of freedom in general relativity – II: Dirac versus Bergmann observables and the objectivity of space-time

Abstract

This is the second of a couple of papers in which we aim to show the peculiar capability of the Hamiltonian ADM formulation of metric gravity to grasp a series of conceptual and technical problems that appear to have not been directly discussed so far. In this paper we also propose new viewpoints about issues that, being deeply rooted into the foundational level of Einstein theory, seem particularly worth of clarification in connection with the alternative programs of string theory and loop quantum gravity. The achievements of the present work include:1) the analysis of the so-called Hole phenomenology in strict connection with the Hamiltonian treatment of the initial value problem. The work is carried through in metric gravity for the class of Christoudoulou-Klainermann space-times, in which the temporal evolution is ruled by the weak ADM energy. It is crucial to our analysis the re-interpretation of active diffeomorphisms as passive and metric-dependent dynamical symmetries of Einstein's equations, a re-interpretation which enables to disclose their (nearly unknown) connection to gauge transformations on-shell; this is expounded in the first paper (gr-qc/0403081).2) the utilization of the Bergmann-Komar intrinsic pseudo-coordinates, defined as suitable functionals of the Weyl curvature scalars, as tools for a peculiar gauge-fixing to the super-hamiltonian and super-momentum constraints;3) the consequent construction of a physical atlas of 4-coordinate systems for the 4-dimensional mathematical manifold, in terms of the highly non-local degrees of freedom of the gravitational field (its four independent Dirac observables). Such construction embodies the physical individuation of the points of space-time as point-events, both in absence and presence of matter, and associates a non-commutative structure to each gauge fixing or 4-dimensional coordinate system. 4) a clarification of the multiple definition given by Peter Bergmann of the concept of (Bergmann) observable in general relativity. This clarification leads to the proposal of a main conjecture asserting the existence of i) special Dirac's observables which are also Bergmann's observables, ii) gauge variables that are coordinate independent (namely they behave like the tetradic scalar fields of the Newman-Penrose formalism). A by-product of this achievements is the falsification of a recently advanced argument asserting the absence of (any kind of) change in the observable quantities of general relativity. 5) a proposal showing how the physical individuation of point-events could in principle be implemented as an experimental setup and protocol leading to a standard of space-time more or less like atomic clocks define standards of time.In the end, against the well-known Einstein's assertion according to which general covariance takes away from space and time the last remnant of physical objectivity, we conclude that pointevents maintain a peculiar sort of objectivity. Also, besides being operationally essential for building measuring apparatuses for...

Moderate structural realism about space-time

“…An approach to this problem based on the original suggestion of Synge (1960) and on the work of Bergmann & Komar (1960) has been recently developed by Lusanna & Pauri (2005) and discussed by Dorato & Pauri (2005). The original Bergmann-Komar procedure aims at providing locally and in the absence of non-gravitational (energy-)matter fields a "pseudocoordinatization" of space-time points from four suitable functions of the four functionally independent eigenvalues of the Weyl tensor (the "trace free part" of the Riemann tensor) (the coordinatization is "pseudo" because it does not define a chart in the atlas of the considered manifold).…”

“…For space-times with no symmetries, this procedure seeks to provide distinct sets of four real numbers for distinct space-time points, which are therefore labelled by these noncoordinate sets of four real numbers. For a specific (although quite large) class of "empty" solutions of the Einstein field equations, Lusanna & Pauri (2005) develop the BergmannKomar procedure in the constrained Hamiltonian framework of GR: roughly speaking, an arbitrary foliation of space-time into three-dimensional hypersurfaces is considered, and the "dynamics" of the theory is given by a Hamiltonian function, provided that the canonical variables satisfy certain constraints. 13 Within this framework, GR can be understood as a constrained Hamiltonian theory, that is, a theory whose physical content is invariant under gauge transformations generated by the first class primary constraints.…”

Hamiltonian expression of curvature tensors in the York canonical basis: (I) Riemann tensor and Ricci scalars

“…whether they are 4-scalars like in the old proposal of Bergmann [5] (the so-called Bergmann observables (BO) of the gravitational field; see also Ref. [6,7]).…”

Quantum astrometric observables: Time delay in classical and quantum gravity

“…5.7 of [2]). Theoretically, the algebra of physical observables, with each observable a mathematical model of an experimental outcome, is an integral part of a complete classical or quantum theory of gravity [3][4][5][6][7][8].…”

“…Second, one has to identify elements of this class that correspond to outcomes of some experiments of interest. The literature on this subject is extensive [3][4][5][6][7][8][9] (see also references therein), but no solution has been entirely successful. To illustrate the difficulties, consider the following two examples.…”