Proofs involving covariant derivatives

This article contains proof of formulas in Riemannian geometry that involve the Christoffel symbols.

## Contents

* 1 Contracted Bianchi identities
* 1.1 Proof
* 2 The covariant divergence of the Einstein tensor vanishes
* 2.1 Proof
* 3 The Lie derivative of the metric
* 3.1 Proof
* 4 See also
* 5 References
* 6 Books

## Contracted Bianchi identities

### Proof

Start with the Bianchi identity[1]

Rabmn;ℓ+Rabℓm;n+Rabnℓ;m=0.

Contract both sides of the above equation with a pair of metric tensors:

gbngam(Rabmn;ℓ+Rabℓm;n+Rabnℓ;m)=0,

gbn(Rmbmn;ℓ−Rmbmℓ;n+Rmbnℓ;m)=0,

gbn(Rbn;ℓ−Rbℓ;n−Rbmnℓ;m)=0,

Rnn;ℓ−Rnℓ;n−Rnmnℓ;m=0.

The first term on the left contracts to yield a Ricci scalar, while the third term contracts to yield a mixed Ricci tensor,

R;ℓ−Rnℓ;n−Rmℓ;m=0.

The last two terms are the same (changing dummy index n to m) and can be combined into a single term which shall be moved to the right,

R;ℓ=2Rmℓ;m,

which is the same as

∇mRmℓ=12∇ℓR.

Swapping the index labels l and m yields

∇ℓRℓm=12∇mR, Q.E.D. (return to article)

## The covariant divergence of the Einstein tensor vanishes

### Proof

The last equation in the proof above can be expressed as

∇ℓRℓm−12δℓm∇ℓR=0

where δ is the Kronecker delta. Since the mixed Kronecker delta is equivalent to the mixed metric tensor,

δℓm=gℓm,

and since the covariant derivative of the metric tensor is zero (so it can be moved in or out of the scope of any such derivative), then

∇ℓRℓm−12∇ℓgℓmR=0.

Factor out the covariant derivative

∇ℓ(Rℓm−12gℓmR)=0,

then raise the index m throughout

∇ℓ(Rℓm−12gℓmR)=0.

The expression in parentheses is the Einstein tensor, so [1]

∇ℓGℓm=0, Q.E.D. (return to article)

this means that the covariant divergence of the Einstein tensor vanishes.

## The Lie derivative of the metric

### Proof

Starting with the local coordinate formula for a covariant symmetric tensor field g=gab(xc)dxa⊗dxb, the Lie derivative along a vector field X=Xa∂a is

ℒXgab=Xc∂cgab+gcb∂aXc+gca∂bXc=Xc∂cgab+gcb(∂aXc±ΓdacXd)+gca(∂bXc±ΓdbcXd)=(Xc∂cgab−gcbΓdacXd−gcaΓdbcXd)+[gcb(∂aXc+ΓdacXd)+gca(∂bXc+ΓdbcXd)]=Xc∇cgab+gcb∇aXc+gca∇bXc=0+gcb∇aXc+gca∇bXc=gcb∇aXc+gca∇bXc=∇aXb+∇bXa

here, the notation ∂a=∂∂xa means taking the partial derivative with respect to the coordinate xa.  Q.E.D. (return to article)

## See also

* Four-gradient
* Palatini identity
* Ricci calculus

## References

1. ↑ 1.0 1.1 Synge J.L., Schild A. (1949). Tensor Calculus. pp. 87–89–90.

## Books

* Bishop, R.L.; Goldberg, S.I. (1968), Tensor Analysis on Manifolds (First Dover 1980 ed.), The Macmillan Company, ISBN 0-486-64039-6, https://archive.org/details/tensoranalysison00bish
* Danielson, Donald A. (2003). Vectors and Tensors in Engineering and Physics (2/e ed.). Westview (Perseus). ISBN 978-0-8133-4080-7.
* Lovelock, David; Rund, Hanno (1989). Tensors, Differential Forms, and Variational Principles. Dover. ISBN 978-0-486-65840-7.
* Synge J.L., Schild A. (1949). Tensor Calculus. first Dover Publications 1978 edition. ISBN 978-0-486-63612-2. https://archive.org/details/tensorcalculus00syng.
* J.R. Tyldesley (1975), An introduction to Tensor Analysis: For Engineers and Applied Scientists, Longman, ISBN 0-582-44355-5
* D.C. Kay (1988), Tensor Calculus, Schaum’s Outlines, McGraw Hill (USA), ISBN 0-07-033484-6
* T. Frankel (2012), The Geometry of Physics (3rd ed.), Cambridge University Press, ISBN 978-1107-602601

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