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S.Carrol Exercise G.10

  1. Nov 7, 2007 #1
    1. The problem statement

    I need to prove that if two metrics are related by an overall conformal transformation of the form \overline{g}_{ab}=e^{a(x)}g_{ab} and if k^{a} is a killing vector for the metric g_{ab} then k^{a} is a conformal killing vector for the metric \overline{g}_{ab}

    2. Relevant equations

    killing equation
    killing conformal equation

    3. The attempt at a solution

    i think i need to show that \overline{\nabla}_{a}k_{b}+\overline{\nabla}_{b}k_{a}=(k^{r}\nabla_{r}a(x))\overline{g}_{ab}

    which as far as i understand is the killing conformal equation for the metric \overline{g}_{ab}

    so using the relation \overline{\nabla}_{a}k_{b}=\nabla_{a}k_{b}-C^{r}_{ab}k_{c}

    where C^{r}_{ab} are the connection coefficients relating the derivative operatrors for g_{ab} and \overline{g}_{ab}

    i sustitute this in \overline{\nabla}_{a}k_{b}+\overline{\nabla}_{b}k_{a}

    and using killing equation for the metric g_{ab} i obtain:


    which is not the conformal killing equation so im lost , can anyone help me on this?
  2. jcsd
  3. Nov 8, 2007 #2


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    Try using LaTex in your post; you may get more of a response. Use the tags [ tex] [ /tex] or [ itex] [ /itex] for normal Tex and inline, respectively (without the spaces in the brackets).
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