Torsion tensor

Let $\nabla$ be an affine connexion on a $𝐶 𝛼$ -manifold $𝑀$ . The torsion tensor $𝑇 𝑐 𝑎 𝑏 \in T 12 (𝑀)$ is a tensor field defined by #m/def/geo/diff

𝑇 𝑐 𝑎 𝑏 𝑋 𝑎 𝑌 𝑏 = 𝑇 (𝑋, 𝑌) 𝑐 = \nabla 𝑋 𝑌 𝑐 - \nabla 𝑌 𝑋 𝑐 - [𝑋, 𝑌] 𝑐 .

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

Proof of tensoriality

By the Leibniz rule,

𝑇 (𝑓 𝑋, 𝑌) 𝑐 = 𝑇 (𝑋, 𝑓 𝑌) 𝑐 = 𝑓 𝑇 (𝑋, 𝑌) 𝑐

so we have a $𝐶 𝛼 (𝑀)$ -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

0 = [\nabla 𝑎, \nabla 𝑏] 𝑓 + 𝑇 𝑐 𝑎 𝑏 \nabla 𝑐 𝑓 .

Proof

Let $𝑋 𝑎, 𝑌 𝑎 \in 𝔛 (𝑀)$ be vector fields and $𝑓 \in 𝐶 𝛼 (𝑀)$ be a scalar field. Then

\nabla 𝑋 \nabla 𝑌 𝑓 = (𝑋 \circ 𝑌) [𝑓] = 𝑋 𝑎 \nabla 𝑎 𝑌 𝑏 \nabla 𝑏 𝑓 = 𝑋 𝑎 𝑌 𝑏 \nabla 𝑎 \nabla 𝑏 + (𝑋 𝑎 \nabla 𝑎 𝑌 𝑏) \nabla 𝑏 𝑓 = 𝑋 𝑎 𝑌 𝑏 \nabla 𝑎 \nabla 𝑏 + \nabla \nabla 𝑋 𝑌 𝑓

and thus

[\nabla 𝑋, \nabla 𝑌] 𝑓 = [𝑋, 𝑌] 𝑐 \nabla 𝑐 𝑓 = 𝑋 𝑎 𝑌 𝑏 [\nabla 𝑎, \nabla 𝑏] 𝑓 + (\nabla 𝑋 𝑌 𝑐 - \nabla 𝑌 𝑋 𝑐) \nabla 𝑐 𝑓

𝑋 𝑎 𝑌 𝑏 [\nabla 𝑎, \nabla 𝑏] 𝑓 = - (\nabla 𝑋 𝑌 𝑐 - \nabla 𝑌 𝑋 𝑐 - [𝑋, 𝑌] 𝑐) \nabla 𝑐 𝑓 = - 𝑋 𝑎 𝑌 𝑏 𝑇 𝑐 𝑎 𝑏 \nabla 𝑐 𝑓

as required.

Properties

$𝑇 𝑐 𝑎 𝑏 = 𝑇 𝑐 [𝑎 𝑏]$ .

Other results

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.

See also the Contorsion tensor and Riemannian curvature

#state/tidy | #lang/en | #SemBr

Torsion tensor

Properties

Other results

Related