1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Conformal Transformation of the metric (General Relativity)

  1. Nov 8, 2007 #1
    1. The problem statement, all variables and given/known data
    I need to prove that if two metrics are related by an overall conformal transformation of the form [tex]\overline{g}_{ab}=e^{a(x)}g_{ab}[/tex] and if [tex]k^{a}[/tex] is a killing vector for the metric [tex]g_{ab}[/tex] then [tex]k^{a}[/tex] is a conformal killing vector for the metric [tex]\overline{g}_{ab}[/tex]

    2. Relevant equations

    killing equation
    killing conformal equation

    3. The attempt at a solution

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

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

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

    where [tex]C^{r}_{ab}[/tex] are the connection coefficients for the conformal transformation, i.e., if [tex]\overline{g}_{ab}=\omega^{2}g_{ab}[/tex] then:

    [tex]C^{r}_{ab}=\omega^{-1}(\delta^{r}_{a}\nabla_{b}\omega+\delta^{r}_{b}\nabla_{a}\omega-g_{ab}g^{rc}\nabla_{c}\omega)[/tex] if i substitute this in [tex]\overline{\nabla}_{a}k_{b}+\overline{\nabla}_{b}k_ {a}[/tex]

    and use killing equation for the metric [tex]g_{ab}[/tex] i obtain:


    which is not the conformal killing equation for [tex]\overline{g}_{ab}[/tex] so im lost , can anyone help me on this?
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you help with the solution or looking for help too?

Similar Discussions: Conformal Transformation of the metric (General Relativity)