Reference frames are crucial for describing local observers in general relativity. In quantum gravity, we have different proposals for how to understand them. There are models in which the reference frames remain classical. In other approaches, they are fundamentally quantum. Recently, two options appeared to investigate these possibilities at the level of the classical and quantum algebra of observables. One option is based on the covariant phase space approach, using gravitational edge modes. In the canonical approach, there is another option, 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. We introduce the corresponding charges and explain how this parameter, which disappears from metric gravity, modifies the generators for angle-dependent asymptotic symmetries. This modification affects the supertranslation charges, while the global charges remain unchanged.

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• General relativity
• Quantum foundations
• Quantum gravity