The Physical Role of Gravitational and Gauge Degrees of Freedom in General Relativity - II: Dirac versus Bergmann observables and the Objectivity of Space-Time

TL;DR

This paper explores the nature of observables in general relativity, clarifies their definitions, and investigates the physical individuation of space-time points through Hamiltonian analysis and gauge fixing.

Contribution

It proposes a main conjecture about special Dirac and Bergmann observables and analyzes the gauge-invariant structure of space-time points using Hamiltonian formalism.

Findings

Clarified Bergmann's multiple definitions of observables.

Proposed existence of special Dirac and Bergmann observables.

Connected active diffeomorphisms to gauge transformations on-shell.

Abstract

(abridged)The achievements of the present work include: a) A clarification of the multiple definition given by Bergmann of the concept of {\it (Bergmann) observable. This clarification leads to the proposal of a {\it 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). b) The analysis of the so-called {\it Hole} phenomenology in strict connection with the Hamiltonian treatment of the initial value problem in metric gravity for the class of Christoudoulou -Klainermann space-times, in which the temporal evolution is ruled by the {\it weak} ADM energy. It is crucial the re-interpretation of {\it active} diffeomorphisms as {\it 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). The use of the Bergmann-Komar {\it intrinsic pseudo-coordinates} allows to construct a {\it physical atlas} of 4-coordinate systems for the 4-dimensional {\it mathematical} manifold, in terms of the highly non-local degrees of freedom of the gravitational field (its four independent {\it Dirac observables}), and to realize the {\it physical individuation} of the points of space-time as {\it point-events} as a gauge-fixing problem, also associating a non-commutative structure to each 4-coordinate system.