Gauge Fixing and Observables in General Relativity

TL;DR

This paper investigates the structure of gauge symmetries in general relativity, clarifies the nature of time translation invariance, and constructs explicit time-dependent invariants using intrinsic coordinates, with illustrations from a relativistic free particle.

Contribution

It demonstrates that time is not frozen in general relativity and constructs explicit time-dependent invariants using intrinsic coordinates, retaining lapse and shift as canonical variables.

Findings

Time translation is not a realized symmetry in phase space.

Time-dependent invariants can be constructed using intrinsic coordinates.

Fluctuations in proper time are demonstrated in the free particle example.

Abstract

The conventional group of four-dimensional diffeomorphisms is not realizeable as a canonical transformation group in phase space. Yet there is a larger field-dependent symmetry transformation group which does faithfully reproduce 4-D diffeomorphism symmetries. Some properties of this group were first explored by Bergmann and Komar. More recently the group has been analyzed from the perspective of projectability under the Legendre map. Time translation is not a realizeable symmetry, and is therefore distinct from diffeomorphism-induced symmetries. This issue is explored further in this paper. It is shown that time is not "frozen". Indeed, time-like diffeomorphism invariants must be time-dependent. Intrinsic coordinates of the type proposed by Bergmann and Komar are used to construct invariants. Lapse and shift variables are retained as canonical variables in this approach, and therefore will be subject to quantum fluctuations in an eventual quantum theory. Concepts and constructions are illustrated using the relativistic classical and quantum free particle. In this example concrete time-dependent invariants are displayed and fluctuation in proper time is manifest.