Connexion disagreement tensor

Let and denote affine connexions on a -manifold . The connexion disagreement tensor of with is a tensor field defined so that for we have¹

and thus

Proof of tensoriality

We need to show that

is a -linear map into . To this end let . Then by the Leibniz rule,

as required.

A word of warning for physicists

Physicists might be uncomfortable with the assertion that is a tensor, and most introductory general relativity courses will spend a lot of time stressing that connexion coëfficients such as the Christoffel symbols are not tensors. Depending on perspective this is either a misunderstanding or disagreement. The connexion coëfficients for a coördinate chart are not covariant since it depended on the choice of coördinate chart, but if you consider the partial derivative as a local affine connexion as extra data attached to our manifold which we retain after change of coördinates, they suddenly are tensorial.

In particular, given local coördinates and considering partial derivative as a local affine connexion, we typically denote the connexion disagreement of an affine connexion with is denoted and we have

or in components

We call the connexion coëfficients. If is the Levi-Civita symbol the connexion coëfficients are called the Christoffel symbols.

Covariant derivative disagreement on vector fields

With the same notation as above, let be a vector field. Then

Proof

Let and . Then since covariant derivatives all agree with the exterior derivative on scalar fields, we have

Therefore by the Leibniz rule

for all . The conclusion follows.

Covariant derivative disagreement on tensor fields

With the same notation as above, let be a tensor field, where we will suppress position since no raising or lowering will take place. Then by induction on applications of the Leibniz rule we see

In other words, to convert from one connexion to the other for the covariant derivative of a general tensor field , we must

add a contraction with each upper index of ,
subtract a contraction with each lower index of .

Other properties

If both and are torsion-free, or more generally if they have the same Contorsion tensor, then is symmetric in its lower indices.

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2009. General relativity, §1.1, pp. 32–33. ↩