How Do Vielbeins Function in Higher-Dimensional Kaluza-Klein Theory?

[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?

 

[DailyMather]

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
$$\begin{align}<br /> \hat{e}^{a}=e^{\alpha\phi}{e}^{a}<br /> \end{align}<br /> \begin{align}<br /> \hat{e}^{z}=e^{\beta\phi}(dz+A_{\mu}dx^{\mu})<br /> \end{align}<br /> \begin{align}<br /> \hat{E}_A^M\hat{e}_M^B=\delta_A^B<br /> \end{align}<br /> where A,B are the flat indexes in 5d and M,N the curved ones. How can I calculate for example <br /> \hat{E}_z^{\mu} from this relation? is it correct to use this relation like in the following expressions?<br /> \begin{align} <br /> \delta_a^z =\hat{e}_a^M\hat{E}_M^z=\hat{e}_a^{\mu}\hat{E}_{\mu}^z+\hat{e}_a^{z}\hat{E}_{z}^{z}<br /> \end{align}$$

Are there any books where I can find more about vielbeins in more than 4d?

Just fixed the latex for you :)