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Vielbein for kaluza-klein

  1. Jan 28, 2010 #1

    cje

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    I try to check the calculation that Kaluza and Klein originally did. It is in the first pages of Pope's lecture notes: http://faculty.physics.tamu.edu/pope/ihplec.pdf
    I have a few problems with the calculation of the Ricci tensor with spin connection, namely with the inverse vielbeins. The vielbeins are given by the following expressions
    \begin{align}
    \hat{e}^{a}=e^{\alpha\phi}{e}^{a}
    \end{align}
    \begin{align}
    \hat{e}^{z}=e^{\beta\phi}(dz+A_{\mu}dx^{\mu})
    \end{align}
    \begin{align}
    \hat{E}_A^M\hat{e}_M^B=\delta_A^B
    \end{align}
    where A,B are the flat indexes in 5d and M,N the curved ones. How can I calculate for example
    $ \hat{E}_z^{\mu}$ from this relation? is it correct to use this relation like in the following expressions?
    \begin{align}
    \delta_a^z =\hat{e}_a^M\hat{E}_M^z=\hat{e}_a^{\mu}\hat{E}_{\mu}^z+\hat{e}_a^{z}\hat{E}_{z}^{z}
    \end{align}
    Are there any books where I can find more about vielbeins in more than 4d?
     
  2. jcsd
  3. Jan 29, 2010 #2
    Just fixed the latex for you :)
     
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