The issue of time in generally covariant theories and the Komar-Bergmann approach to observables in general relativity

TL;DR

This paper explores the distinction between time evolution and symmetry transformations in generally covariant theories, emphasizing that time-dependent invariants can be constructed via gauge fixing, with implications for observables in general relativity.

Contribution

It demonstrates that gauge invariants in general relativity are inherently time-dependent and cannot be constants of motion, contrasting with simpler models like the relativistic free particle.

Findings

Time-dependent invariants arise through gauge fixing.

Gauge invariants in general relativity are not constants of motion.

Explicit examples using Weyl curvature scalars are provided.

Abstract

Diffeomorphism-induced symmetry transformations and time evolution are distinct operations in generally covariant theories formulated in phase space. Time is not frozen. Diffeomorphism invariants are consequently not necessarily constants of the motion. Time-dependent invariants arise through the choice of an intrinsic time, or equivalently through the imposition of time-dependent gauge fixation conditions. One example of such a time-dependent gauge fixing is the Komar-Bergmann use of Weyl curvature scalars in general relativity. An analogous gauge fixing is also imposed for the relativistic free particle and the resulting complete set time-dependent invariants for this exactly solvable model are displayed. In contrast with the free particle case, we show that gauge invariants that are simultaneously constants of motion cannot exist in general relativity. They vary with intrinsic time.