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

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

Just fixed the latex for you :)

 

[Entropia]

The use of vielbeins is essential in the Kaluza-Klein theory to account for the extra dimension. The expressions provided for the vielbeins are correct, but your confusion seems to be with the calculation of $\hat{E}_z^{\mu}$. In order to calculate this, we can use the relation $\delta_a^z = \hat{e}_a^M \hat{E}_M^z$, as you have correctly identified. However, we must be careful in how we interpret this relation. The indices $a$ and $z$ refer to the flat five-dimensional space, while the indices $M$ and $\mu$ refer to the curved four-dimensional spacetime. Therefore, we cannot simply equate the two expressions as you have done in the last line. Instead, we must use the fact that $\hat{e}_a^{\mu} = e_a^{\mu}$ and $\hat{e}_a^z = A_a$, as given in the expressions for the vielbeins. This allows us to rewrite the relation as $\delta_a^z = e_a^{\mu} \hat{E}_{\mu}^z + A_a \hat{E}_z^z$. From this, we can solve for $\hat{E}_z^{\mu}$ by isolating it on one side of the equation. This calculation is typically done in a more systematic way using the Cartan structure equations, but this approach should also yield the correct result.

As for resources on vielbeins in higher dimensions, there are several books and articles that discuss this topic. Some good references include "Gauge Fields, Knots and Gravity" by Baez and Muniain, "Gauge Theories in Particle Physics" by Aitchison and Hey, and "Modern Kaluza-Klein Theories" by Overduin and Wesson. Additionally, there are many research papers and lecture notes available online that discuss the use of vielbeins in higher dimensions. With some further research, you should be able to find more resources that can help you with your calculations and understanding of this topic.