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

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.