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Homework Statement:: Please see below.
Relevant Equations:: Please see below.
I am trying to find a reference to a textbook or a paper that details the following time-invariance Kaluza-Klein metric:
\begin{equation}
\mathrm{d}s^2_{(5)}=\lambda_{ab}\left(\mathrm{d}x^a+\omega^{a}_{\phantom{a}i}\mathrm{d}x^i\right)(\mathrm{d}x^b+\omega^{b}_{\phantom{b}j}\mathrm{d}x^j)+\frac{1}{\tau}h_{ij}\mathrm{d}x^i\mathrm{d}x^j.
\end{equation}
So far, I can only find sources for the Kaluza-Klein metric but not it's time-invariant version. Can someone point me to a relevant reference?
Relevant Equations:: Please see below.
I am trying to find a reference to a textbook or a paper that details the following time-invariance Kaluza-Klein metric:
\begin{equation}
\mathrm{d}s^2_{(5)}=\lambda_{ab}\left(\mathrm{d}x^a+\omega^{a}_{\phantom{a}i}\mathrm{d}x^i\right)(\mathrm{d}x^b+\omega^{b}_{\phantom{b}j}\mathrm{d}x^j)+\frac{1}{\tau}h_{ij}\mathrm{d}x^i\mathrm{d}x^j.
\end{equation}
So far, I can only find sources for the Kaluza-Klein metric but not it's time-invariant version. Can someone point me to a relevant reference?