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Derivation of the Christoffel symbol 
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#1
Jan2909, 11:07 AM

P: 1

How can I derive the Christoffel symbol from the vanishing of the covariant derivative of the metric tensor? can somebody write the calculation, I read that I have to do some permutation and resumming but I don't get the result! Thank you!



#2
Jan2909, 06:40 PM

Emeritus
Sci Advisor
PF Gold
P: 9,403

Did you obtain a result like
[tex]g_{\rho\sigma,\mu}=\Gamma^\lambda_{\mu\rho}g_{\lambda\sigma}+\Gamma^\la mbda_{\mu\sigma}g_{\rho\lambda}[/tex]already? In that case, consider the quantity [tex]g_{\mu\sigma,\rho}+g_{\mu\rho,\sigma}g_{\rho\sigma,\mu}[/tex]and I think you'll be able to figure out the rest. Don't forget that a LeviCivita connection is torsion free. This implies that the Christoffel symbol is symmetric in the lower indices. 


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