Derivation of the Christoffel symbol

  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. jcsd
  3. Fredrik

    Fredrik 10,314
    Staff Emeritus
    Science Advisor
    Gold Member

    Did you obtain a result like

    [tex]g_{\rho\sigma,\mu}=\Gamma^\lambda_{\mu\rho}g_{\lambda\sigma}+\Gamma^\lambda_{\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 Levi-Civita connection is torsion free. This implies that the Christoffel symbol is symmetric in the lower indices.
     
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