Geometric relational framework for general-relativistic gauge field theories

TL;DR

This paper develops a relational framework for general-relativistic gauge theories using an advanced dressing field method, leading to relational Einstein equations that incorporate scalar coordinatization concepts.

Contribution

It introduces a novel formulation based on the automorphisms of principal bundles, extending the dressing field method to a relational setting for local field theories.

Findings

Relational Einstein equations derived from the framework.

Implementation of the generalised point-coincidence argument.

Unification of scalar coordinatization approaches in GR.

Abstract

We remind how relationality arises as the core insight of general-relativistic gauge field theories from the articulation of the generalised hole and point-coincidence arguments. Hence, a compelling case for a manifestly relational framework ensues naturally. We propose our formulation for such a framework, based on a significant development of the dressing field method of symmetry reduction. We first develop a version for the group $\text{Aut}(P)$ of automorphisms of a principal bundle $P$ over a manifold $M$, as it is the most natural and elegant, and as $P$ hosts all the mathematical structures relevant to general-relativistic gauge field theory. Yet, as the standard formulation is local, on $M$, we then develop the relational framework for local field theory. It manifestly implements the generalised point-coincidence argument, whereby the physical field-theoretical degrees of freedoms co-define each other and define, coordinatise, the physical spacetime itself. Applying the framework to General Relativity, we obtain relational Einstein equations, encompassing various notions of "scalar coordinatisation" \`a la Kretschmann-Komar and Brown-Kucha\v{r}.