# Vielbein for kaluza-klein

1. Jan 28, 2010

### cje

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. Jan 29, 2010

### DailyMather

Just fixed the latex for you :)