What is a reference frame in General Relativity?

TL;DR

The paper tackles the challenge of defining local, gauge-invariant observables in General Relativity (GR) by reframing reference frames as physical systems instantiated by four scalar degrees of freedom that establish a local diffeomorphism, enabling relational localization of the metric via $g_{IJ}(\phi)=[(\phi^{(I)})^{-1}]^* g_{ab}$. It introduces a three-fold taxonomy—Idealised Reference Frames (IRFs), Dynamical Reference Frames (DRFs), and Real Reference Frames (RRFs)—and connects these to the construction of Dirac observables and the interpretation of diffeomorphism gauge freedom, with DRFs exemplified by four Klein-Gordon fields and the GPS frame. The work shows that IRFs provide relational localization but do not yield genuine gauge-invariant observables, whereas DRFs and RRFs integrate frame dynamics (and backreaction where applicable) to produce complete observables and a physically meaningful gauge interpretation, effectively deparametrising GR in relational terms. By clarifying the coordinate-frame distinction and illustrating practical implications for measurement modelling, the paper lays a conceptual groundwork for both classical and quantum gravity contexts, including the emergence of quantum reference frames and their backreaction on spacetime. Overall, the framework advances a relational, physically instantiated account of reference frames that addresses long-standing issues in GR and offers a pathway toward extending these ideas in quantum gravitational settings.

Abstract

This work introduces a novel three-fold classification of reference frames in General Relativity, distinguishing between Idealised Reference Frames (IRFs), Dynamical Reference Frames (DRFs), and Real Reference Frames (RRFs). By defining a reference frame as a set of degrees of freedom instantiated by a physical system, the work contrasts this notion with that of coordinate systems-purely mathematical idealisations lacking physical instantiation. This classification addresses two longstanding challenges in GR: (P1) the difficulty of defining local and gauge-invariant observables, and (P2) how to interpret diffeomorphism gauge freedom in physical terms rather than as merely a mathematical redundancy. Overall, this work clarifies the conceptual foundations in classical General Relativity, enhancing our understanding of gauge-symmetries, observers and laying the groundwork for future investigations in both classical and quantum gravitational contexts.

Figures (1)

• Figure 1: Construction of the set of GPS reference frame. Each scalar $\phi^{(I)}$ corresponds to the proper time of a satellite broadcast at point $P$.