Now I'm just start to study the Kaluza-Klein theory from http://arxiv.org/abs/grqc/9805018.(adsbygoogle = window.adsbygoogle || []).push({});

I try to calculate the Einstein Field equations in 5 dimensional vacuum space-time.

we start with 5D metric tensor,

[tex]

\hat{g}_{AB}=\begin{pmatrix}

g_{\alpha}_{\beta}+k^{2}\phi^{2}A_{\alpha}A_{\beta} & k\phi^{2}A_{\alpha}\\

k\phi^{2}A_{\beta}&\phi^{2}

\end{pmatrix}[/tex]

where A,B indices run from 0,1,2,3,4 and [tex]\alpha,\beta[/tex] run from 0,1,2,3

Next, I have to calculate Christoffel connection from:

[tex]\displaystyle \hat{\Gamma}^{A}_{BC}=\frac{1}{2}\hat{g}^{AD}(\partial_{B}{\hat{g}_{CD}}+\partial_{C}{\hat{g}_{BD}}-\partial_{D}{\hat{g}_{BC}})[/tex]

Kaluza propose the extra condition called "cylindrical condition" that says all derivative involve with the fifth coordinate must vanish. so we can conclude that

[tex]

\partial_4{\hat{g}_{AB}}= \partial_4{\hat{\Gamma}^{A}_{BC}}=0

[/tex]

then all the connection would be

[tex]

\displaystyle\hat{\Gamma}^{4}_{44},\hat{\Gamma}^{4}_{4\alpha},\hat{\Gamma}^{\alpha}_{44},\hat{\Gamma}^{\alpha}_{4\beta},\hat{\Gamma}^{4}_{\alpha\beta},\hat{\Gamma}^{\alpha}_{\beta\gamma}

[/tex]

Here's is my questions, I confuse about the calculation of the last two connection [tex]\hat{\Gamma}^{4}_{\alpha\beta},\hat{\Gamma}^{\alpha}_{\beta\gamma}[/tex]

for an example:

[tex]

\displaystyle \hat{\Gamma}^{\sigma}_{\alpha\beta}=\frac{1}{2}\hat{g}^{\sigma D}(\partial_{\alpha}{\hat{g}_{\beta D}}+\partial_{\beta}{\hat{g}_{\alpha D}}-\partial_{D}{\hat{g}_{\alpha\beta}})

[/tex]

I'm confused about index sum "D". Can I just replace index D with 4 dimensional (Greek)index or I have to sum it to 4D index plus the fifth-D ones.

[tex]

\displaystyle \hat{\Gamma}^{\sigma}_{\alpha\beta}=\frac{1}{2}\hat{g}^{\sigma D}(\partial_{\alpha}{\hat{g}_{\beta D}}+\partial_{\beta}{\hat{g}_{\alpha D}}-\partial_{D}{\hat{g}_{\alpha\beta}})=\frac{1}{2}[\hat{g}^{\sigma \lambda}(\partial_{\alpha}{\hat{g}_{\beta \lambda}}+\partial_{\beta}{\hat{g}_{\alpha\lambda}}-\partial_{\lambda}{\hat{g}_{\alpha\beta}})+\hat{g}^{\sigma 4}(\partial_{\alpha}{\hat{g}_{\beta 4}}+\partial_{\beta}{\hat{g}_{\alpha 4}})

[/tex]

or just

[tex]\displaystyle \hat{\Gamma}^{\sigma}_{\alpha\beta}=\frac{1}{2}\hat{g}^{\sigma\lambda}(\partial_{\alpha}{\hat{g}_{\beta \lambda}}+\partial_{\beta}{\hat{g}_{\alpha\lambda}}-\partial_{\lambda}{\hat{g}_{\alpha\beta}})[/tex]

which one is correct?

otherwise you may be just told me a book or paper that the explicit form of all possible connections in Kaluza-Klein theory were expressed. I would be appreciate.

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Christoffel connection in Kaluza Klein Theory

**Physics Forums | Science Articles, Homework Help, Discussion**