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]
Contract both sides of the above equation with a pair of metric tensors:
The first term on the left contracts to yield a Ricci scalar, while the third term contracts to yield a mixed Ricci tensor,
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,
which is the same as
Swapping the index labels l and m yields
The covariant divergence of the Einstein tensor vanishes
Proof
The last equation in the proof above can be expressed as
where δ is the Kronecker delta. Since the mixed Kronecker delta is equivalent to the mixed metric tensor,
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
Factor out the covariant derivative
then raise the index m throughout
The expression in parentheses is the Einstein tensor, so [1]
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 , the Lie derivative along a vector field is
![{\displaystyle {\begin{aligned}{\mathcal {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}{\bigl (}\partial _{a}X^{c}\pm \Gamma _{da}^{c}X^{d}{\bigr )}+g_{ca}{\bigl (}\partial _{b}X^{c}\pm \Gamma _{db}^{c}X^{d}{\bigr )}\\&={\bigl (}X^{c}\partial _{c}g_{ab}-g_{cb}\Gamma _{da}^{c}X^{d}-g_{ca}\Gamma _{db}^{c}X^{d}{\bigr )}+{\bigl [}g_{cb}{\bigl (}\partial _{a}X^{c}+\Gamma _{da}^{c}X^{d}{\bigr )}+g_{ca}{\bigl (}\partial _{b}X^{c}+\Gamma _{db}^{c}X^{d}{\bigr )}{\bigr ]}\\&=X^{c}\nabla _{c}g_{ab}+g_{cb}\nabla _{a}X^{c}+g_{ca}\nabla _{b}X^{c}\\&=0+g_{cb}\nabla _{a}X^{c}+g_{ca}\nabla _{b}X^{c}\\&=g_{cb}\nabla _{a}X^{c}+g_{ca}\nabla _{b}X^{c}\\&=\nabla _{a}X_{b}+\nabla _{b}X_{a}\end{aligned}}}](../I/m/3b47e81ee4f393d2e39f9e701dce31a59557fa69.svg)
here, the notation means taking the partial derivative with respect to the coordinate . Q.E.D. (return to article)
See also
References
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; Rund, Hanno (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