S-Wave for minimally coupled scalar field

Question

This question is in reference to the paper here (Equation 3).The extremal 3-brane metic in $D=10$ can be written as: \begin{equation*} ds^2 = A^{-1/2}(-dt^2 +dx_1^2 +dx^2+ dx^3) + A^{1/2}(dr^2 +r^2 d\Omega_5^2) \end{equation*} where \begin{equation*} A = 1+ \frac{R^4}{r^4} \end{equation*} In this background the $s$-wave of a minimally coupled massless scalar satisfies: \begin{equation*} \left[\rho^{-5}\frac{d}{d\rho}\rho^{5}\frac{d}{d\rho}+ \frac{(\omega R)^{4}}{\rho^{4}}+1\right]\phi(\rho) =0 \end{equation*} How do I derive this result?

This post imported from StackExchange Physics at 2014-07-28 11:15 (UCT), posted by SE-user Debangshu

Comments

Please consider making your question self-contained

This post imported from StackExchange Physics at 2014-07-28 11:16 (UCT), posted by SE-user user1504

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Quick guess, haven't tried it: 1. Work out the Klein-Gordon equation for $\phi$ using the standard expressions for the curved space Laplacian. 2. Make an s-wave ansatz. 3. Simplify.

This post imported from StackExchange Physics at 2014-07-28 11:16 (UCT), posted by SE-user Michael Brown