Help with Kaluza Klein Christoffel symbols

[user1139]

Homework Statement

I want to calculate $\tilde{\Gamma}^\lambda_{\mu 5}$.

Relevant Equations

\begin{align}
\tilde{\Gamma}^\lambda_{\mu\nu} & = \frac{1}{2} \tilde{g}^{\lambda X} \left(\partial_\mu \tilde{g}_{\nu X} + \partial_\nu \tilde{g}_{\mu X} - \partial_X \tilde{g}_{\mu\nu}\right) \\
& =\frac{1}{2} \tilde{g}^{\lambda\sigma} \left(\partial_\mu \tilde{g}_{\nu\sigma} + \partial_\nu \tilde{g}_{\mu\sigma} - \partial_\sigma \tilde{g}_{\mu\nu}\right) + \frac{1}{2} \tilde{g}^{\lambda5} \left(\partial_\mu \tilde{g}_{\nu5} + \partial_\nu \tilde{g}_{\mu 5} - \partial_5 \tilde{g}_{\mu\nu}\right)
\end{align}

where

\begin{cases}
\tilde{g}_{\mu\nu} = g_{\mu\nu} + k A_\mu A_\nu \\
\tilde{g}_{\mu5} = k A_\mu \\
\tilde{g}_{55} = k\,(\mathrm{constant})
\end{cases}

and

\begin{cases}
\tilde{g}^{\mu\nu} = g^{\mu\nu} \\
\tilde{g}^{\mu5} = -A_\mu \\
\tilde{g}^{55} = \frac{1}{k} + A_\mu A^\mu.
\end{cases}

If I want to calculate $\tilde{\Gamma}^\lambda_{\mu 5}$, I will write

\begin{align}
\tilde{\Gamma}^\lambda_{\mu 5} & = \frac{1}{2} \tilde{g}^{\lambda X} \left(\partial_\mu \tilde{g}_{5 X} + \partial_5 \tilde{g}_{\mu X} - \partial_X \tilde{g}_{\mu 5}\right) \\
& =\frac{1}{2} \tilde{g}^{\lambda\sigma} \left(\partial_\mu \tilde{g}_{5\sigma} + \partial_5 \tilde{g}_{\mu\sigma} - \partial_\sigma \tilde{g}_{\mu 5}\right) + \frac{1}{2} \tilde{g}^{\lambda5} \left(\partial_\mu \tilde{g}_{55} + \partial_5 \tilde{g}_{\mu5} - \partial_5 \tilde{g}_{\mu 5}\right)
\end{align}

Is it then correct to write that the above reduces to

$$\tilde{\Gamma}^\lambda_{\mu 5}=\frac{1}{2} \tilde{g}^{\lambda\sigma} \left(\partial_\mu \tilde{g}_{5\sigma} - \partial_\sigma \tilde{g}_{\mu 5}\right)?$$

 

[anuttarasammyak]

Do you mean all the indexes take number of 0,1,2,3,4,5 though usually 0,1,2,3?

 

[Dr Transport]

Kaluza-Klein is a 5-dimensional theory, if I remember my Relativity correctly.