[Paper Review] Lost in translation: the Abelian affine connection (in the coincident gauge)

Jose Beltrán Jiménez, Tomi Koivisto|arXiv (Cornell University)|Feb 3, 2022

Cosmology and Gravitation TheoriesPhysics and Astronomy57 references47 citations

TL;DR

This paper clarifies the use of the coincident gauge in symmetric teleparallel gravity, where the Abelian affine connection is flat and torsion-free, allowing the connection to be locally set to zero via coordinate redefinition. The key contribution is exposing persistent affine symmetries and resolving misconceptions about gauge freedom, physical content, and matter coupling in this formalism.

ABSTRACT

The simplest i.e. the Abelian i.e. the commutative i.e. the integrable i.e. the flat and torsion-free i.e. the symmetric teleparallel affine connection has been considered in many recent works in the literature. Such an affine connection is characterised by the property that it can be vanished by a general coordinate transformation, by fixing the so called coincident gauge. This article focuses on the subtleties involved in the applications of the coincident gauge.

Motivation & Objective

• To resolve widespread misconceptions and persistent myths in the literature regarding the coincident gauge in symmetric teleparallel gravity.
• To clarify the physical versus mathematical role of the coincident gauge, distinguishing it from physical symmetries.
• To demonstrate that the coincident gauge does not eliminate physical content, even though it trivializes the connection.
• To analyze the residual affine symmetries and their field-theoretic implications in the gauge-fixed framework.
• To show that matter coupling in this formalism excludes second-clock effects and related anomalies, generalizing to arbitrary symmetric connections.

Proposed method

• Uses differential geometry to show that a symmetric teleparallel connection is locally equivalent to the trivial connection via coordinate transformation, defining the coincident gauge.
• Applies the concept of Stueckelbergisation (or Kretschmannisation) to reframe general covariance as a physical symmetry, not a formal redundancy.
• Analyzes the residual affine symmetry ξα → Mαβξβ + ξα0 after gauge fixing, showing it persists and underlies physical content.
• Derives the connection form as Γαμν = ∂ξα/∂ξλ ∂μ∂νξλ, showing its integrability and flatness via [∇μ, ∇ν] = 0.
• Examines matter coupling in the coincident gauge, proving the absence of second-clock effects due to the symmetric connection structure.
• Compares the formalism to metric teleparallelism, drawing parallels to clarify confusions about 'good' vs. 'bad' tetrads in related frameworks.

Experimental results

Research questions

• RQ1 What is the true physical role of the coincident gauge in symmetric teleparallel gravity, and why is it often misunderstood?
• RQ2 How do residual affine symmetries manifest in the coincident gauge, and what is their physical significance?
• RQ3 Why does matter coupling in this framework exclude second-clock effects, and how does this generalize to arbitrary symmetric connections?
• RQ4 How does the Stueckelbergisation procedure clarify the physical content of general covariance in this context?
• RQ5 What are the geometric and field-theoretic consequences of fixing the connection to zero via coordinate choice?

Key findings

• The coincident gauge is a valid and physically consistent choice that trivializes the connection via coordinate redefinition, but it does not eliminate physical content.
• The residual affine symmetry ξα → Mαβξβ + ξα0 persists after gauge fixing and is responsible for the physical degrees of freedom in the theory.
• The connection is flat and torsion-free, ensuring integrability via [∇μ, ∇ν] = 0, which characterizes it as the gauge connection of a translation gauge theory.
• Matter coupling in this framework excludes second-clock effects, as shown by the absence of non-minimal couplings to the connection in the symmetric case.
• The coincident gauge does not obstruct the analysis of symmetric solutions such as spherically symmetric or cosmological spacetimes.
• The formalism generalizes to arbitrary symmetric connections, and the absence of second-clock effects holds universally in such theories.