1. Limited time only! Sign up for a free 30min personal 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!

Homework Help: Weyl Tensor invariant under conformal transformations

  1. Mar 16, 2016 #1
    1. The problem statement, all variables and given/known data
    As the title says, I need to show this. A conformal transformation is made by changing the metric:
    ##g_{\mu\nu}\mapsto\omega(x)^{2}g_{\mu\nu}=\tilde{g}_{\mu\nu}##

    2. Relevant equations
    The Weyl tensor is given in four dimensions as:
    ##
    C_{\rho\sigma\mu\nu}=R_{\rho\sigma\mu\nu}-\left(g_{\rho\left[\mu\right.}R_{\left.v\right]\sigma}-g_{\sigma\left[\mu\right.}R_{\left.v\right]\rho}\right)+\frac{1}{3}g_{\rho\left[\mu\right.}g_{\left.\nu\right]\sigma}R
    ##

    where ##R_{\mu\nu}## is the Ricci tensor, ##R## is the Ricci scalar, and ##R_{\rho\sigma\mu\nu}## is the Riemann tensor
    3. The attempt at a solution
    ##
    \begin{eqnarray*}
    \tilde{g}_{\rho\left[\mu\right.}R_{\left.v\right]\sigma}&=&\frac{1}{2}\left(\tilde{g}_{\rho\mu}R_{\nu\sigma}-\tilde{g}_{\rho\nu}R_{\mu\sigma}\right)=\frac{1}{2}\omega(x)^{2}\left(g_{\rho\mu}R_{\nu\sigma}-g_{\rho\nu}R_{\mu\sigma}\right)\\\tilde{g}_{\sigma\left[\mu\right.}R_{\left.v\right]\rho}&=&\frac{1}{2}\left(\tilde{g}_{\sigma\mu}R_{\nu\rho}-\tilde{g}_{\sigma\nu}R_{\mu\rho}\right)=\frac{1}{2}\omega(x)^{2}\left(g_{\sigma\mu}R_{\nu\rho}-g_{\sigma\nu}R_{\mu\rho}\right)\\\tilde{g}_{\rho\left[\mu\right.}\tilde{g}_{\left.\nu\right]\sigma}&=&\frac{1}{2}\left(\tilde{g}_{\rho\mu}\tilde{g}_{\nu\sigma}-\tilde{g}_{\rho\nu}\tilde{g}_{\mu\sigma}\right)=\frac{1}{2}\omega(x)^{4}\left(g_{\rho\mu}g_{\nu\sigma}-g_{\rho\nu}g_{\mu\sigma}\right)
    \end{eqnarray*}
    ##

    From here, I am lost. How do I make the ##\omega(x)## vanish?
     
  2. jcsd
  3. Mar 16, 2016 #2

    Twigg

    User Avatar
    Gold Member

    If ## g_{\mu\nu} \to \omega (x)^{2}g_{\mu\nu}##, what about ##\Gamma_{\mu\nu}^{\sigma}##? Is ##\tilde{R}_{\rho\sigma\mu\nu}## the same as ##R_{\rho\sigma\mu\nu}##?
     
  4. Mar 17, 2016 #3
    Thanks! That gave me the push in the right direction! Managed to solve it now.
     
  5. May 21, 2018 #4
    Sorry to dig an old post, but I’m currently struggling with Weyl tensor conformal invariance as well.

    I started with the following assumptions:

    - the invariant tensor is not ##C_{abcd}## but ##C^a\,_{bcd}##
    - Connection was not metrically compatible
    - the transformation I considered was slightly different but basically equivalent: ##g_{\mu\nu}=e^{-2\omega}g_{\mu\nu}##

    In this case, invariance was immediate as neither ##\Gamma^\mu_{\nu\rho}##, nor the Riemann or the Ricci change under conformal rescaling, but only the scalar curvature and the Ricci with one index up (##R^\mu\,_\nu\equiv g^{\mu\lambda}R_{\lambda\nu}##).

    Now, I was trying to prove conformal invariance with metrical connection, so with nontrivial modifications of connection coefficients, Riemann, Ricci tensor and scalar: as I’m stuck with huge formulas, could anyone confirm if it’s normal or if I’m missing some simplifying argument?
    I’m not asking for detailed calculations but feel free to post them if you want.

    Thanks!
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted