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,

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

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