Lost in translation: the Abelian affine connection (in the coincident gauge)

TL;DR

This paper discusses the subtleties involved in applying the coincident gauge to the Abelian affine connection, which is flat, torsion-free, and can be transformed away by coordinate changes.

Contribution

It clarifies the conceptual and practical issues in using the coincident gauge for the Abelian affine connection in teleparallel gravity theories.

Findings

Highlights the limitations of the coincident gauge approach

Identifies potential pitfalls in the application of the gauge

Provides insights into the geometric interpretation of the connection

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.