Torsion tensor

Let be an affine connexion on a -manifold . The torsion tensor is a tensor field defined by diff

A connexion for which the torsion tensor vanishes is said to be torsion-free.

Proof of tensoriality

By the Leibniz rule,

so we have a -bilinear map, and therefore a tensor field.

We can interpret the torsion tensor as measuring the extent to which covariant derivatives fail to commute on scalar fields, is the sense that

Proof

Let be vector fields and be a scalar field. Then

and thus

so

as required.

Properties

Exterior derivative from a torsion-free connexion
The idea that at an isolated point we cannot detect gravity is equivalent to the connexion on spacetime being torsion free, i.e. find local inertial reference frames. This also implies the equivalence of inertial and gravitational mass.