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Let us quickly run through the standard KK compactification. We start with a $d+1$ dimensional theory $$S = \frac{1}{16\pi G_{d+1}} \int d^{d+1}x \sqrt{G} R_{d+1}$$ More general actions on the $d+1$ dimensional space can be considered, but this will suffice for our purposes. The metric $G_{MN}$ can be decomposed as $$ds^2 = G_{MN} dx^M dx^N = e^{2\Phi} ... 2 The way we measure length is to use the metric tensor. Any spacetime has a metric tensor associated with it, and it's the metric tensor that is responsible for the notion of distance. To make this a little less abstract consider a concrete example. In flat spacetime the metric tensor is just:$$ ds^2 = -c^2dt^2 + dx^2 + dy^2 + dx^2  Suppose you want to ...