torsion tensor

A solder form on a vector bundle allows one to define the torsion tensor of a connection.

The first expression is called the torsion tensor of the connection, and the second is also called the curvature.

A torsion tensor in differential geometry.

In differential geometry, an affine connection is torsion-free if its torsion tensor vanishes.

In the resulting field equations, the torsion tensor is a homogeneous, linear function of the spin tensor.

The Ricci tensor is no longer symmetric because it contains the nonzero torsion tensor.

The torsion tensor can be decomposed into two irreducible parts: a trace-free part and another part which contains the trace terms.

An affine connection is 'determined by' its family of affinely parameterized geodesics, up to torsion tensor .

In particular (see below) while the geodesic equations determine the symmetric part of the connection, the torsion tensor determines the antisymmetric part.

In one version of SUGRA (but certainly not the only one), we have the following constraints upon the torsion tensor:

(Also, if there is nonzero torsion, the first Bianchi identity becomes a differential identity of the torsion tensor.)

Therefore, metric-affine gravitation theory possesses a different collection of dynamic variables which are a pseudo-Riemannian metric, a non-metricity tensor and a torsion tensor.

This theory removes a constraint of the symmetry of the affine connection and treats its antisymmetric part, the torsion tensor, as a variable in varying the action.

Connections also lead to convenient formulations of geometric invariants, such as the curvature (see also curvature tensor and curvature form), and torsion tensor.

New teleparallel gravity theory (or new general relativity) is a theory of gravitation on Weitzenböck space-time, and attributes gravitation to the torsion tensor formed of the parallel vector fields.

In some alternative theories like Einstein-Cartan theory, the stress-energy tensor may not be perfectly symmetric because of a nonzero spin tensor, which geometrically corresponds to a nonzero torsion tensor.

The reason for the non-obvious sum in the definition is that the contortion tensor, being the difference between two metric-compatible Christoffel symbols, must be antisymmetric in the last two indices, whilst the torsion tensor itself is antisymmetric in its first two indices.

There are actually many different versions of SUGRA out there which are inequivalent in the sense that their actions and constraints upon the torsion tensor are different, but ultimately equivalent in that we can always perform a field redefinition of the supervierbeins and spin connection to get from one version to another.