Geometric Relational Framework for General-Relativistic Gauge Field Theories

Abstract

It is recalled how relationality arises as the core insight of general-relativistic gauge field theories from the articulation of the generalized hole and point-coincidence arguments. Hence, a compelling case for a manifestly relational framework ensues naturally. A formulation for such a framework is proposed, based on a significant development of the dressing field method of symmetry reduction. A version for the group $Aut\left(P\right)$ of automorphisms of a principal bundle $P$ over a manifold $M$ is first developed, as it is the most natural and elegant, and as $P$ hosts all the mathematical structures relevant to general-relativistic gauge field theory. However, as the standard formulation is local, on $M$, the relational framework for local field theory is then developed. The generalized point-coincidence argument is manifestly implemented, whereby the physical field-theoretical degrees of freedoms co-define each other and define, coordinatize, the physical spacetime itself. Applying the framework to General Relativity, relational Einstein equations are obtained, encompassing various notions of “scalar coordinatization” à la Kretschmann–Komar and Brown–Kuchař.