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I am trying to work through the calculations from the following article.

I need to find the s-wave radial wave equation for a minimally coupled scalar field, given by $\partial_{\mu}( \sqrt {-g} g^{\mu \nu} \partial_{\nu}\phi ) = 0$, for the metric of a non-extremal black 3-brane:

$ ds^{2} = H^{-1/2}(r) [-f(r) dt^{2} + d\mathbf{x}^{2}] + H^{1/2}(r)[f^{-1}(r) dr^{2} + r^{2}d\Omega_{5}^{2}]$

where $H(r)= 1 + \frac{R^{4}}{r^{4}}$ and $f(r) = 1 - \frac{r_{0}^{4}}{r^{4}}$. The result to be obtained is the following:

$\phi '' + \frac{5r^{4} - r_{0}^{4}}{r(r^{4} - r_{0}^{4})}\phi ' + \omega^{2} \frac{r^{4}(r^{4} + R^{4})}{(r^{4} - r_{0}^{4})^{2}} \phi = 0$

I am a bit confused by the dependence of $g_{\mu\nu}$ on $r$ and am not sure how to get to this result. Could anyone help me a bit?

Thanks in advance.

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