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Showing that the Einstein Tensor has zero divergence

  1. Apr 26, 2013 #1
    1. The problem statement, all variables and given/known data
    We have

    [tex]R_{iklm;n}+R_{iknl;m}+R_{ikmn;l} \equiv 0[/tex]

    Show that by multiplying above with [itex]g^{im}g^{kn}[/itex]

    we'll get

    [tex]\left( R^{ik}-\frac{1}{2} g^{ik} R \right)_{;k}[/tex]

    2. The attempt at a solution

    [tex]g^{im}g^{kn} \left( R_{iklm;n}+R_{iknl;m}+R_{ikmn;l} \right) \equiv 0[/tex]

    [tex]g^{im}R_{i} ^{n}_{lm;n}+g^{kn}R^{m}_{knl;m}+\frac{\partial R}{\partial x^l} \equiv 0 [/tex]

    [tex]R^{n}_{l;n}+R^{m}_{l;m}+\frac{\partial R}{\partial x^l} \equiv 0[/tex]

    then I'm stuck, not sure how to proceed. Honestly I'm not sure if my contractions are correct either. Please help?
  2. jcsd
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