Geometric classification of the torsion tensor in space-time

TL;DR

This paper introduces two classification schemes for torsion tensors in space-time, based on their geometric construction and irreducible decomposition, and explores their implications in cosmological models and physical phenomena.

Contribution

It proposes novel classification methods for torsion tensors and applies them to analyze their effects on energy-momentum and kinematic quantities in cosmology.

Findings

Classified torsion tensors using bivector-vector products and irreducible decomposition.

Analyzed how torsion sources modify energy-momentum tensors.

Studied torsion's impact on shear, vorticity, expansion, and acceleration.

Abstract

Torsion appears in literature in quite different forms. Generally, spin is considered to be the source of torsion, but there are several other possibilities in which torsion emerges in different contexts. In some cases a phenomenological counterpart is absent, in some other cases torsion arises from sources without spin as a gradient of a scalar field. Accordingly, we propose two classification schemes. The first one is based on the possibility to construct torsion tensors from the product of a covariant bivector and a vector and their respective space-time properties. The second one is obtained by starting from the decomposition of torsion into three irreducible pieces. Their space-time properties again lead to a complete classification. The classifications found are given in a U_4, a four dimensional space-time where the torsion tensors have some peculiar properties. The irreducible decomposition is useful since most of the phenomenological work done for torsion concerns four dimensional cosmological models. In the second part of the paper two applications of these classification schemes are given. The modifications of energy-momentum tensors are considered that arise due to different sources of torsion. Furthermore, we analyze the contributions of torsion to shear, vorticity, expansion and acceleration. Finally the generalized Raychaudhuri equation is discussed.