Affine connexion

Connexion disagreement tensor

Let ∇ and ˜∇ denote affine connexions on a 𝐶𝛼-manifold 𝑀. The connexion disagreement tensor 𝐶𝑐𝑎𝑏 ∈T12(𝑀) of ∇ with ˜∇ is a tensor field defined so that for 𝜔𝑎 ∈Ω1(𝑀) we have¹

𝐶𝑐𝑎𝑏𝜔𝑐:=(˜∇𝑎−∇𝑎)𝜔𝑏

and thus

∇𝑎𝜔𝑏=˜∇𝑎𝜔𝑏−𝐶𝑐𝑎𝑏𝜔𝑐.

Proof of tensoriality

We need to show that

𝐶(𝜔)𝑎𝑏:=𝐶𝑐𝑎𝑏𝜔𝑐=(˜∇𝑎−∇𝑎)𝜔𝑏

is a 𝐶𝛼(𝑀)-linear map into T02(𝑀). To this end let 𝑓 ∈𝐶𝛼(𝑀). Then by the Leibniz rule,

𝐶(𝑓𝜔)𝑎𝑏=(˜∇𝑎−∇𝑎)(𝑓𝜔)𝑏=d𝑓𝑎𝜔𝑏+𝑓˜∇𝑎𝜔𝑏−d𝑓𝑎𝜔𝑏=𝑓(˜∇𝑎−∇𝑎)𝜔𝑏=𝑓𝐶(𝜔)𝑎𝑏

as required.

A word of warning for physicists

Physicists might be uncomfortable with the assertion that 𝐶𝑐𝑎𝑏 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 𝑥 :𝑈 →ℝ𝑚 and considering partial derivative as a local affine connexion, we typically denote the connexion disagreement of an affine connexion ∇ with 𝜕 is denoted Γ𝑐𝑎𝑏 and we have

∇𝑎𝜔𝑏=𝜕𝑎𝜔𝑏−Γ𝑐𝑎𝑏𝜔𝑐.

or in components

∇𝜇𝜔𝜈=𝜕𝜇𝜔𝜈−Γ𝛿𝜇𝜈𝜔𝛿.

We call Γ𝛿𝜇𝜈 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 𝑋𝑎 ∈𝔛(𝑀) be a vector field. Then

∇𝑎𝑋𝑏=˜∇𝑎𝑋𝑏+𝐶𝑏𝑎𝑐𝑋𝑐.

Proof

Let 𝑋𝑎 ∈𝔛(𝑀) and 𝜔𝑎 ∈Ω1(𝑀). Then since covariant derivatives all agree with the exterior derivative on scalar fields, we have

(˜∇𝑎−∇𝑎)(𝜔𝑏𝑋𝑏)=0.

Therefore by the Leibniz rule

0=𝜔𝑏˜∇𝑎𝑋𝑏+𝑋𝑏˜∇𝑎𝜔𝑏−𝜔𝑏∇𝑎𝑋𝑏−𝑋𝑏∇𝑎𝜔𝑏=𝜔𝑏(˜∇𝑎−∇𝑎)𝑋𝑏+𝑋𝑏(˜∇𝑎−∇𝑎)𝜔𝑏=𝜔𝑏(˜∇𝑎−∇𝑎)𝑋𝑏+𝑋𝑏𝐶𝑐𝑎𝑏𝜔𝑐=𝜔𝑏((˜∇𝑎−∇𝑎)𝑋𝑏+𝐶𝑏𝑎𝑐𝑋𝑐)

for all 𝜔𝑎 ∈Ω1(𝑀). The conclusion follows.

Covariant derivative disagreement on tensor fields

With the same notation as above, let 𝑇𝑎1⋯𝑎𝑝𝑏1⋯𝑏𝑞 ∈T𝑝𝑞(𝑀) 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

∇𝑐𝑇𝑎1⋯𝑎𝑝𝑏1⋯𝑏𝑞=˜∇𝑐𝑇𝑎1⋯𝑎𝑝𝑏1⋯𝑏𝑞+𝑝∑𝑖=1𝐶𝑎𝑖𝑐𝑑𝑇𝑎1⋯𝑑⋯𝑎𝑝𝑏1⋯𝑏𝑞−𝑞∑𝑖=1𝐶𝑑𝑐𝑏𝑖𝑇𝑎1⋯𝑎𝑝𝑏1⋯𝑑⋯𝑏𝑞.

In other words, to convert from one connexion to the other for the covariant derivative of a general tensor field 𝑇𝑎1⋯𝑎𝑝𝑏1⋯𝑏𝑞, we must

• add a contraction with each upper index of 𝑇𝑎1⋯𝑎𝑝𝑏1⋯𝑏𝑞,
• subtract a contraction with each lower index of 𝑇𝑎1⋯𝑎𝑝𝑏1⋯𝑏𝑞.

Other properties

1. If both ∇ and ˜∇ are torsion-free, or more generally if they have the same Contorsion tensor, then 𝐶𝑐𝑎𝑏 =𝐶𝑐(𝑎𝑏) is symmetric in its lower indices.

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Footnotes

1. 2009. General relativity, §1.1, pp. 32–33. ↩