Proofs involving covariant derivatives

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

Contracted Bianchi identities

Proof

Start with the Bianchi identity^[1]

$R_{{abmn;l}}+R_{{ablm;n}}+R_{{abnl;m}}=0\,\!$.

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

$g^{{bn}}g^{{am}}(R_{{abmn;l}}+R_{{ablm;n}}+R_{{abnl;m}})=0,\,\!$

$g^{{bn}}(R^{m}{}_{{bmn;l}}-R^{m}{}_{{bml;n}}+R^{m}{}_{{bnl;m}})=0,\,\!$

$g^{{bn}}(R_{{bn;l}}-R_{{bl;n}}-R_{b}{}^{m}{}_{{nl;m}})=0,\,\!$

$R^{n}{}_{{n;l}}-R^{n}{}_{{l;n}}-R^{{nm}}{}_{{nl;m}}=0.\,\!$

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

$R_{{;l}}-R^{n}{}_{{l;n}}-R^{m}{}_{{l;m}}=0.\,\!$

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,

$R_{{;l}}=2R^{m}{}_{{l;m}},\,\!$

which is the same as

$\nabla _{m}R^{m}{}_{l}={1 \over 2}\nabla _{l}R\,\!$.

Swapping the index labels l and m yields

$\nabla _{l}R^{l}{}_{m}={1 \over 2}\nabla _{m}R\,\!$,      Q.E.D.     (return to article)

The covariant divergence of the Einstein tensor vanishes

Proof

The last equation in Proof 1 above can be expressed as

$\nabla _{l}R^{l}{}_{m}-{1 \over 2}\delta ^{l}{}_{m}\nabla _{l}R=0\,\!$

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

$\delta ^{l}{}_{m}=g^{l}{}_{m},\,\!$

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

$\nabla _{l}R^{l}{}_{m}-{1 \over 2}\nabla _{l}g^{l}{}_{m}R=0.\,\!$

Factor out the covariant derivative

$\nabla _{l}\left(R^{l}{}_{m}-{1 \over 2}g^{l}{}_{m}R\right)=0,\,\!$

then raise the index m throughout

$\nabla _{l}\left(R^{{lm}}-{1 \over 2}g^{{lm}}R\right)=0.\,\!$

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

$\nabla _{l}G^{{lm}}=0,\,\!$     Q.E.D.    (return to article)

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

See also

• Four-gradient
• Ricci calculus

References

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
• Danielson, Donald A. (2003). Vectors and Tensors in Engineering and Physics (2/e ed.). Westview (Perseus). ISBN 978-0-8133-4080-7.
• Lovelock, David; Hanno Rund (1989) [1975]. 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.
• 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