Observables in classical canonical gravity: folklore demystified

TL;DR

This paper clarifies conceptual misunderstandings in classical canonical gravity, demonstrating that proper terminology resolves paradoxes and enabling the construction of gauge-invariant, time-evolving observables using scalar coordinatization.

Contribution

It provides a thorough analysis of gauge invariance issues and introduces a method to construct observables in covariant theories via scalar coordinatization, resolving longstanding paradoxes.

Findings

All paradoxes disappear with correct terminology.

Time evolving observables can be constructed for every observer.

Scalar coordinatization is essential for gauge fixing and observable construction.

Abstract

We give an overview of some conceptual difficulties, sometimes called paradoxes, that have puzzled for years the physical interpetation of classical canonical gravity and, by extension, the canonical formulation of generally covariant theories. We identify these difficulties as stemming form some terminological misunderstandings as to what is meant by "gauge invariance", or what is understood classically by a "physical state". We make a thorough analysis of the issue and show that all purported paradoxes disappear when the right terminology is in place. Since this issue is connected with the search of observables - gauge invariant quantities - for these theories, we formally show that time evolving observables can be constructed for every observer. This construction relies on the fixation of the gauge freedom of diffeomorphism invariance by means of a scalar coordinatization. We stress the condition that the coordinatization must be made with scalars. As an example of our method for obtaining observables we discuss the case of the massive particle in AdS spacetime.