# Flat spin connections in the Teleparallel equivalent of General Relativity

## TL;DR

This paper explores the role of flat spin connections in the teleparallel equivalent of general relativity, showing their decoupling from tetrads and matter fields, and analyzing implications for modified gravity theories.

## Contribution

It generalizes the treatment of flat spin connections, revealing their structure as a subspace and their influence on torsion and inertial effects without affecting matter dynamics.

## Key findings

- Flat spin connections form a subspace in the affine space of spin connections.
- Translations in this subspace introduce torsion related to inertial effects.
- Matter field dynamics remain invariant under changes in flat spin connections.

## Abstract

This work generalizes the treatment of flat spin connections in the teleparallel equivalent of general relativity. It is shown that a general flat spin connection form a subspace in the affine space of spin connections which is dynamically decoupled from the tetrad and the matter fields. A translation in the affine subspace introduces a torsion term without changing the tetrad. Instead, the change in the torsion is related to the introduction of a global acceleration field term that introduces Lorentz inertial effects in the reference frame. The dynamics of the gravitationally coupled matter fields remains however equivalent regardless of the flat spin connection chosen. The implications of the break of this invariance by a general $f(T)$ and $f(R)$ is discussed.