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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.