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Christoffel Symbol Question

  1. Apr 24, 2012 #1
    Is the following true?
    [tex]\Gamma_{\mu \nu \alpha}+\Gamma_{\nu \mu \alpha}=-2\Gamma_{\alpha \mu \nu}[/tex]
    where:
    [tex]\Gamma_{\alpha \mu \nu}=g_{\alpha \sigma}\Gamma^{\sigma}_{~\mu \nu}[/tex]

    I ask because, while bored in a philosophy lecture, I decided to try to derive the geodesic equation by extremizing ∫gμνuμuνdλ, where uμ = dxμ/dλ.

    I was able to arrive at the following, where aμ=duμ/dλ:
    [tex]2a_\alpha = (\Gamma_{\mu \nu \alpha}+\Gamma_{\nu \mu \alpha})u^\mu u^\nu [/tex]

    So, am I on the right track or did I make an error somewhere?
     
  2. jcsd
  3. Apr 24, 2012 #2
    Nevermind, they are clearly not equal. From the definition of the Christoffel symbols in terms of the metric, I found that:

    [tex](\Gamma_{\mu \nu \alpha}+\Gamma_{\nu \mu \alpha})=\partial_\alpha
    g_{\mu \nu }[/tex]
    This makes sense, because [itex]\nabla_\alpha g_{\mu \nu }=0[/itex].

    Unfortunately for me though, this is clearly not equal to [itex]-2\Gamma_{\alpha \mu \nu}[/itex] given that:

    [tex]-2\Gamma_{\alpha \mu \nu}=\partial_\alpha g_{\mu \nu}-\partial_\mu g_{\nu \alpha}-\partial_\nu g_{\mu \alpha }[/tex]
     
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