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Christoffel connection in Kaluza Klein Theory

  1. Apr 7, 2010 #1
    Now I'm just start to study the Kaluza-Klein theory from http://arxiv.org/abs/grqc/9805018.
    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.
     
    Last edited: Apr 8, 2010
  2. jcsd
  3. Apr 7, 2010 #2

    Mentz114

    User Avatar
    Gold Member

    If the EM field is going to act on a charged test body, then there must be some Christoffel symbols involving the 5th dimension. So I'd guess your first conjecture is right.

    Have a look at this paper. It has a bit on the kinematics of K-K theory ( but that is not it's main interest ).
     

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