Christoffel symbols/Proofs

From Wikipedia, the free encyclopedia

You have new messages (last change).
This mathematics article is devoted entirely to providing mathematical proofs and support for claims and statements made in the article List of formulas in Riemannian geometry. This article is currently an experimental vehicle to see how well we can provide proofs and details for a math article without cluttering up the main article itself. See Wikipedia:WikiProject Mathematics/Proofs for some current discussion. This article is "experimental" in the sense that it is a test of one way we may be able to incorporate more detailed proofs in Wikipedia.

[edit] Proof 1

Start with the Bianchi identity

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)


[edit] Proof 2

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 (R^l {}_m - {1 \over 2} g^l {}_m R) = 0,\,\!

then raise the index m throughout

\nabla_l (R^{lm} - {1 \over 2} g^{lm} R) = 0.\,\!

The expression in parentheses is the Einstein tensor, so

\nabla_l G^{lm} = 0,\,\!     Q.E.D.    (return to article)
Retrieved from "http://en.wikipedia.org../../../c/h/r/Christoffel_symbols_Proofs_57eb.html"