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Geometric relational framework for general-relativistic gauge field theories

Jordan François, Lucrezia Ravera

TL;DR

This work advances a manifestly relational reformulation of general-relativistic gauge theories (gRGFT) by deploying the dressing field method (DFM) within the geometry of field space. It treats the global field space as an infinite-dimensional principal bundle with automorphism group Aut(P), and introduces an associated bundle of regions to encode spacetime integration relationally; dressing fields systematically produce basic, gauge-invariant observables and dressed regions. The key result is a relational set of field equations, notably E^{(⋅)} = 0, that are covariant under field-dependent diffeomorphisms and gauge transformations, thereby making spacetime and internal d.o.f. relationally defined without fixed backgrounds. The framework unifies GR and gauge field theory, clarifies boundary issues, and provides a pathway toward relational quantization, with explicit examples including GR with scalar matter and Einstein–C Maxwell models illustrating scalar coordinatisation and relational Einstein equations.

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" à la Kretschmann-Komar and Brown-Kuchař.

Geometric relational framework for general-relativistic gauge field theories

TL;DR

This work advances a manifestly relational reformulation of general-relativistic gauge theories (gRGFT) by deploying the dressing field method (DFM) within the geometry of field space. It treats the global field space as an infinite-dimensional principal bundle with automorphism group Aut(P), and introduces an associated bundle of regions to encode spacetime integration relationally; dressing fields systematically produce basic, gauge-invariant observables and dressed regions. The key result is a relational set of field equations, notably E^{(⋅)} = 0, that are covariant under field-dependent diffeomorphisms and gauge transformations, thereby making spacetime and internal d.o.f. relationally defined without fixed backgrounds. The framework unifies GR and gauge field theory, clarifies boundary issues, and provides a pathway toward relational quantization, with explicit examples including GR with scalar matter and Einstein–C Maxwell models illustrating scalar coordinatisation and relational Einstein equations.

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 of automorphisms of a principal bundle over a manifold , as it is the most natural and elegant, and as hosts all the mathematical structures relevant to general-relativistic gauge field theory. Yet, as the standard formulation is local, on , 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" à la Kretschmann-Komar and Brown-Kuchař.
Paper Structure (60 sections, 3 theorems, 452 equations, 2 figures)

This paper contains 60 sections, 3 theorems, 452 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Relationality in general-relativistic gauge field theory
  3. General-relativistic physics
  4. Gauge field physics
  5. General-relativistic gauge field theory
  6. Geometry of global field space
  7. Field space as a principal bundle
  8. Natural transformation groups
  9. Differential structure
  10. Tangent bundle and subbundles
  11. Differential forms and their derivations
  12. Remarkable forms
  13. Vertical transformations and gauge transformations
  14. Connections on field space
  15. Ehresmann connections
  16. ...and 45 more sections

Key Result

Proposition 1

If $\ \boldsymbol u : \Phi \ \rightarrow\ \mathcal{D} r[Q, P]$, with $Q\subseteq P$, is s.t. then $\boldsymbol\alpha^{\boldsymbol u} \in \Omega_{\text{tens}}(\Phi, \rho )$ and $\boldsymbol \omega^{\boldsymbol u} \in \mathcal{C}$. Then, their residual $C^\infty(\Phi; \mathop{\mathrm{Aut}}\nolimits_{\text{\scriptsize r}}(P))$-transformations are

Figures (2)

  • Figure 1: Relationality in the general-relativistic framework.
  • Figure 2: Relationality in general-relativistic gauge field theory.

Theorems & Definitions (3)

  • Proposition 1
  • Proposition 2
  • Proposition 3