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Prove that two transformation laws of the Christoffel symbols are the same

  1. Oct 19, 2011 #1
    1. The problem statement, all variables and given/known data

    Prove that the transformation law

    [itex]\Gamma^{\sigma '}_{\lambda '\rho '}=\frac{\partial x^\nu}{\partial x^{\lambda '}}\frac{\partial x^\rho}{\partial x^{\rho '}}\frac{\partial x^{\sigma '}}{\partial x^{\mu}}\Gamma^{\mu}_{\nu\rho}+\frac{\partial x^{\sigma '}}{\partial x^{\mu}}\frac{\partial^2 x^\mu}{\partial x^{\lambda '}\partial x^{\rho '}}[/itex]

    is equivalent to

    [itex]\Gamma^{\sigma '}_{\lambda '\rho '}=\frac{\partial x^\lambda}{\partial x^{\lambda '}}\frac{\partial x^\rho}{\partial x^{\rho '}}\frac{\partial x^{\sigma '}}{\partial x^{\sigma}}\Gamma^{\sigma}_{\lambda\rho}-\frac{ \partial x^\mu}{\partial x^{\lambda '}}\frac{\partial x^{\lambda}}{\partial x^{\rho '}}\frac{\partial^2 x^{\sigma '}}{\partial x^{\mu}\partial x^{\lambda}}[/itex]

    3. The attempt at a solution

    The first term is easy, just relabel the dummy indices [itex] \nu \rightarrow \lambda [/itex] and [itex] \mu \rightarrow \sigma [/itex]. But for the rest of the problem, I have no clue what to do.
     
    Last edited: Oct 19, 2011
  2. jcsd
  3. Oct 21, 2011 #2

    George Jones

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    Maybe this is too late. It's a trick. Differentiate the far left and right of

    [tex]\delta^{\lambda '}_{\mu '} = \frac{\partial x^{\lambda '}}{\partial x^{\mu '}} = \frac{\partial x^{\lambda '}}{\partial x^\rho} \frac{\partial x^\rho}{\partial x^{\mu '}}[/tex]

    with respect to [itex]x^{\alpha '}[/itex].
     
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