Linking Edge Modes and Geometrical Clocks in Linearized Gravity

TL;DR

This paper explores the relationship between edge modes and geometrical clocks in linearized gravity, extending phase space methods and analyzing the impact of the Barbero-Immirzi parameter on asymptotic symmetries.

Contribution

It introduces an extended phase space approach to relate covariant gauge fixing and geometrical clocks in linearized gravity, including effects of the Barbero-Immirzi parameter.

Findings

Gauge fixing conditions recover geometrical clocks in Ashtekar-Barbero variables.

Barbero-Immirzi parameter influences angle-dependent asymptotic charges.

Extended phase space clarifies the relation between edge modes and clocks.

Abstract

Reference frames are crucial for describing local observers in general relativity. In quantum gravity, different proposals exist for how to treat reference frames. There are models with either classical or quantum reference frames. Recently, different choices appeared for investigating these possibilities at the level of the classical and quantum algebra of observables. One choice is based on the covariant phase space approach, using gravitational edge modes. In the canonical approach, there is another choice, relational clocks, built from matter or geometry itself. In this work, we extend existing results and show how to relate edge modes and geometrical clocks in linearized gravity. We proceed in three steps. First, we introduce an extension of the ADM (Arnowitt-Deser-Misner) phase space to account for covariant gauge fixing conditions and the explicit time dependence they add to Hamilton's equations. Second, we show how these gauge fixing conditions recover a specific choice of geometrical clocks in terms of Ashtekar-Barbero connection variables. Third, we study the effect of the Barbero-Immirzi parameter on the generators of asymptotic symmetries and the corresponding charges. This parameter, which disappears from metric gravity, affects the generators for angle-dependent asymptotic symmetries and the corresponding super-translation charges, while it has no effect on the global charges.