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Problem with tensor equation in Cartan formalism

  1. Sep 8, 2014 #1
    I have some problems verifying a particular equation in the Cartan formalism of GR. Basically I want to prove the following equation in three dimensions

    $$\epsilon_{abc} e^a\wedge R^{bc}=\sqrt{|g|}Rd^3 x$$

    where R is the Ricci scalar and $R^{bc}$ is the Ricci curvature

    Attempt at a solution:

    $$\epsilon_{abc} e^a\wedge R^{bc}=\epsilon_{abc} e_\mu^ae_\alpha^be_\beta^c R^{\alpha\beta}_{\nu\rho} dx^\mu\wedge dx^\nu\wedge dx^\rho$$

    Now the idea is that the number of dimensions and the Levi-Civita tensor and the antisymmetry of the three-form forces the set {alpha,beta}={nu,rho}. This will give the expression

    \begin{align}\epsilon_{abc} e^a\wedge R^{bc}&=\epsilon_{abc} e_0^ae_1^be_2^c R^{12}_{12} dx^0\wedge dx^1\wedge dx^2+\\&\epsilon_{abc} e_0^ae_1^be_2^c R^{12}_{21} dx^0\wedge dx^2\wedge dx^1+\\&\epsilon_{abc} e_0^ae_2^be_1^c R^{21}_{21} dx^0\wedge dx^2\wedge dx^1+\\&\epsilon_{abc} e_0^ae_2^be_1^c R^{21}_{12} dx^0\wedge dx^1\wedge dx^2+({\rm cyclic\,permutations})\end{align}

    The problem now is that the Ricci scalar is $$R^{12}_{12}+R^{21}_{21}+({\rm cyclic\,permutations})$$, so when counting the number of terms I obtain $$2\sqrt{|g|}Rd^3 x$$ which is wrong by a factor of 2. Can anyone see where I made a mistake?

    (Maybe this post belongs in the Homework section since the question is definitely homework-like, but since it is NOT homework and the rules/guidelines do not say anything about "homework-like questions", I wasnt sure...)
     
  2. jcsd
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