Teleparallel geometries not characterized by their scalar polynomial torsion invariants

TL;DR

This paper demonstrates that scalar polynomial torsion invariants do not uniquely characterize teleparallel geometries, providing explicit examples and identifying geometries with vanishing invariants.

Contribution

It constructs the most general class of four-dimensional teleparallel geometries not characterized by scalar polynomial torsion invariants.

Findings

Scalar polynomial torsion invariants do not fully characterize teleparallel geometries.

The paper identifies all geometries with vanishing scalar polynomial torsion invariants.

It provides explicit examples of non-uniqueness in teleparallel geometry characterization.

Abstract

A teleparallel geometry is an n-dimensional manifold equipped with a frame basis and an independent spin connection. For such a geometry, the curvature tensor vanishes and the torsion tensor is non-zero. A straightforward approach to characterizing teleparallel geometries is to compute scalar polynomial invariants constructed from the torsion tensor and its covariant derivatives. An open question has been whether the set of all scalar polynomial torsion invariants, $\mathcal{I}_T$ uniquely characterize a given teleparallel geometry. In this paper we show that the answer is no and construct the most general class of teleparallel geometries in four dimensions which cannot be characterized by $\mathcal{I}_T$. As a corollary we determine all teleparallel geometries which have vanishing scalar polynomial torsion invariants.