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Tensors help

  1. Jun 15, 2009 #1

    trv

    User Avatar

    A little stuck while working through a derivation. Hope someone can help.

    1. The problem statement, all variables and given/known data

    Starting from

    [itex]
    -\xi^c(\Gamma^d_{ca}g_{db}+\Gamma^d_{cb}g_{ad})+\partial_b\xi^dg_{ad}+\partial_a\xi^cg_{cb}=0
    [/itex]

    I need to obtain the Killing equations, i.e.

    [itex]
    \bigtriangledown_b\xi_a+\bigtriangledown_a\xi_b=0
    [/itex]

    2. Relevant equations

    3. The attempt at a solution

    Working backwards...

    Rewriting the covariant derivative in terms of the partial derivative gives

    [itex]
    \bigtriangledown_b\xi_a+\bigtriangledown_a\xi_b=\partial_a\xi_b+\partial_b\xi_a-\Gamma^c_{ba}\xi_c-\Gamma^c_{ab}\xi_c=0
    [/itex]

    Lowering the vector in the partial derivatives gives...

    [itex]
    \partial_a\xi_b+\partial_b\xi_a-\Gamma^c_{ba}\xi_c-\Gamma^c_{ab}\xi_c=-\Gamma^c_{ba}\xi_c-\Gamma^c_{ab}\xi_c+\partial_b\xi^dg_{ad}+\partial_a\xi^cg_{cb}=0
    [/itex]

    I don't however know how to go from

    [itex]
    -\Gamma^c_{ba}\xi_c-\Gamma^c_{ab}\xi_c[/itex]

    to

    [itex]
    -\xi^c(\Gamma^d_{ca}g_{db}+\Gamma^d_{cb}g_{ad}[/itex])

    Can someone help?
     
  2. jcsd
  3. Jun 16, 2009 #2
    its a little difficult to show. first you should replace Xi with Xi*metric, then use this metric to lower the index on Gamma, then replace this Gamma with Gamma*metric, which is what we want. hopefully that makes some sense.
     
  4. Jun 17, 2009 #3

    trv

    User Avatar

    Thanks, it does make sense.

    [itex]
    \xi^c\Gamma^d_{ca}g_{bd}=\xi_eg^{ce}\Gamma^d_{ca}g_{bd}=\xi_eg^{ce}\Gamma_{bca}=\xi_e\Gamma^e_{ba}=\xi_c\Gamma^c_{ba}
    [/itex]
     
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