Riemann-Cartan Connection and its Decomposition. One More Assessment of "ECE Theory"

TL;DR

This paper clarifies the symmetries of the Riemann-Cartan connection and contorsion tensor, addressing misconceptions in the literature and providing detailed decomposition and symmetry analysis relevant to ECE theory.

Contribution

It provides a detailed clarification of the symmetries of the contorsion tensor and its decomposition, addressing confusion in the context of ECE theory.

Findings

Contorsion tensor has symmetric and antisymmetric parts.

Symmetric part defines the strain tensor of the connection.

Contorsion tensor exhibits a 'bastard' antisymmetry in covariant indices.

Abstract

In this short pedagogical note we clarify some subtleties concerning the symmetries of the coefficients of a Riemann-Cartan connection and the symmetries of the coefficients of the contorsion tensor that has been a source of some confusion in the literature, in particular in a so called 'ECE theory'. We show in details that the coefficients of the contorsion tensor of a Riemann-Cartan connection has a symmetric part and an antisymmetric part, the symmetric part defining the strain tensor of the connection. Moreover, the contorsion tensor has also a bastard anti-symmetry when written with all its indices in the `covariant' positions.