Proofs involving covariant derivatives

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

Start with the Bianchi identity^[1]

${\displaystyle R_{abmn;\ell }+R_{ab\ell m;n}+R_{abn\ell ;m}=0.}$

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

${\displaystyle g^{bn}{g}^{am}(R_{abmn;\ell }+R_{ab\ell m;n}+R_{abn\ell ;m})=0,}$

${\displaystyle g^{bn}(R^m{}_{bmn;\ell }-R^m{}_{bm\ell ;n}+R^m{}_{bn\ell ;m})=0,}$

${\displaystyle g^{bn}(R_{bn;\ell }-R_{b\ell ;n}-R_b{}^m{}_{n\ell ;m})=0,}$

${\displaystyle R^n{}_{n;\ell }-R^n{}_{\ell ;n}-R^{nm}{}_{n\ell ;m}=0.}$

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

${\displaystyle R_{;\ell }-R^n{}_{\ell ;n}-R^m{}_{\ell ;m}=0.}$

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,

${\displaystyle R_{;\ell }=2R^m{}_{\ell ;m},}$

which is the same as

${\displaystyle {\mathrm{\nabla }}_m{R}^m{}_{\ell }=\frac{1}{2}{\mathrm{\nabla }}_{\ell }R.}$

Swapping the index labels l and m yields

${\displaystyle {\mathrm{\nabla }}_{\ell }{R}^{\ell }{}_m=\frac{1}{2}{\mathrm{\nabla }}_{m}R,}$      Q.E.D.     (return to article)

The last equation in the proof above can be expressed as

${\displaystyle {\mathrm{\nabla }}_{\ell }{R}^{\ell }{}_m-\frac{1}{2}{\delta }^{\ell }{}_m{\mathrm{\nabla }}_{\ell }R=0}$

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

${\displaystyle {\delta }^{\ell }{}_m=g^{\ell }{}_m,}$

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

${\displaystyle {\mathrm{\nabla }}_{\ell }{R}^{\ell }{}_m-\frac{1}{2}{\mathrm{\nabla }}_{\ell }{g}^{\ell }{}_{m}R=0.}$

Factor out the covariant derivative

${\displaystyle {\mathrm{\nabla }}_{\ell }(R^{\ell }{}_m-\frac{1}{2}{g}^{\ell }{}_{m}R)=0,}$

then raise the index m throughout

${\displaystyle {\mathrm{\nabla }}_{\ell }(R^{\ell m}-\frac{1}{2}{g}^{\ell m}R)=0.}$

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

${\displaystyle {\mathrm{\nabla }}_{\ell }{G}^{\ell m}=0,}$     Q.E.D.    (return to article)

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

Starting with the local coordinate formula for a covariant symmetric tensor field ${\displaystyle g=g_{ab}(x^c)d{x}^a\otimes d{x}^b}$, the Lie derivative along a vector field ${\displaystyle X=X^a{\partial }_a}$ is

${\displaystyle \begin{array}{rl}{{ℒ}}_X{g}_{ab}& =X^c{\partial }_c{g}_{ab}+g_{cb}{\partial }_a{X}^c+g_{ca}{\partial }_b{X}^c\\ & =X^c{\partial }_c{g}_{ab}+g_{cb}({\partial }_a{X}^c\pm {\mathrm{\Gamma }}_{da}^c{X}^d)+g_{ca}({\partial }_b{X}^c\pm {\mathrm{\Gamma }}_{db}^c{X}^d)\\ & =(X^c{\partial }_c{g}_{ab}-g_{cb}{\mathrm{\Gamma }}_{da}^c{X}^d-g_{ca}{\mathrm{\Gamma }}_{db}^c{X}^d)+\left[g_{cb}\right({\partial }_a{X}^c+{\mathrm{\Gamma }}_{da}^c{X}^d)+g_{ca}({\partial }_b{X}^c+{\mathrm{\Gamma }}_{db}^c{X}^d\left)\right]\\ & =X^c{\mathrm{\nabla }}_c{g}_{ab}+g_{cb}{\mathrm{\nabla }}_a{X}^c+g_{ca}{\mathrm{\nabla }}_b{X}^c\\ & =0+g_{cb}{\mathrm{\nabla }}_a{X}^c+g_{ca}{\mathrm{\nabla }}_b{X}^c\\ & =g_{cb}{\mathrm{\nabla }}_a{X}^c+g_{ca}{\mathrm{\nabla }}_b{X}^c\\ & ={\mathrm{\nabla }}_a{X}_b+{\mathrm{\nabla }}_b{X}_a\end{array}}$

here, the notation ${\displaystyle {\partial }_a=\frac{\partial }{\partial x^a}}$ means taking the partial derivative with respect to the coordinate ${\displaystyle x^a}$.      Q.E.D.     (return to article)

• Four-gradient
• Palatini identity
• Ricci calculus

• 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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