I don't know why you need AdS/CTF correspondence for that? There is a good derivation found here (page 64 to 90). To give a quick overview:

You can calculate the propagation of gravitational waves on a background metric using pertubation theory in first order. You start with a pertubation of the metric tensor given by: \begin{equation} \widetilde{g}_{\mu\nu} =g_{\mu\nu}+\delta g_{\mu\nu} \qquad\text{with}\qquad |\delta g_{\mu\nu}|<1 \end{equation} \begin{equation} \widetilde{g}^{\mu\nu} =g^{\mu\nu}-\delta g^{\mu\nu} \qquad\text{with}\qquad |\delta g_{\mu\nu}|>1. \end{equation} The different signs were explained by me here already. You can now calculate the Christoffel symbols, the Riemann curvature tensor and the Einstein tensor only in first order of the pertubation.

In a vacuum, we get the equation $G_{\mu\nu}[g]=0$ (equivalent to $R_{\mu\nu}[g]=0$) for the background metric and $\delta G_{\mu\nu}=0$ for the gravitational waves out of the field equations. Using a corresponding pertubation tensor (similar to the correspondence between the Ricci and Einstein tensor): \begin{equation} \delta g_{\mu\nu}' =\delta g_{\mu\nu}-\frac{1}{2}(g^{\kappa\lambda}\delta g_{\kappa\lambda})g_{\mu\nu}, \end{equation} \begin{equation} \delta g_{\mu\nu} =\delta g_{\mu\nu}'-\frac{1}{2}(g^{\kappa\lambda}\delta g_{\kappa\lambda}')g_{\mu\nu}, \end{equation} the equation describing the scattering of gravitational waves is given by: \begin{equation} \square\delta g_{\mu\nu}' +2R_{\kappa\mu\lambda\nu}\delta{g^{\kappa\lambda}}'=0. \end{equation} For the Schwarzschild metric in particular using this equation results in the Regge-Wheeler equation.

I don't know why you need AdS/CTF correspondence for that? There is a good derivation found here (page 64 to 90). To give a quick overview:

You can calculate the propagation of gravitational waves on a background metric using pertubation theory in first order. You start with a pertubation of the metric tensor given by: \begin{equation} \widetilde{g}_{\mu\nu} =g_{\mu\nu}+\delta g_{\mu\nu} \qquad\text{with}\qquad |\delta g_{\mu\nu}|<1 \end{equation} \begin{equation} \widetilde{g}^{\mu\nu} =g^{\mu\nu}-\delta g^{\mu\nu} \qquad\text{with}\qquad |\delta g_{\mu\nu}|>1. \end{equation} The different signs were explained by me here already. You can now calculate the Christoffel symbols, the Riemann curvature tensor and the Einstein tensor only in first order of the pertubation.

In a vacuum, we get the equation $G_{\mu\nu}[g]=0$ (equivalent to $R_{\mu\nu}[g]=0$) for the background metric and $\delta G_{\mu\nu}=0$ for the gravitational waves out of the field equations. Using a corresponding pertubation tensor (similar to the correspondence between the Ricci and Einstein tensor): \begin{equation} \delta g_{\mu\nu}' =\delta g_{\mu\nu}-\frac{1}{2}(g^{\kappa\lambda}\delta g_{\kappa\lambda})g_{\mu\nu}, \end{equation} \begin{equation} \delta g_{\mu\nu} =\delta g_{\mu\nu}'-\frac{1}{2}(g^{\kappa\lambda}\delta g_{\kappa\lambda}')g_{\mu\nu}, \end{equation} the equation describing the scattering of gravitational waves is given by: \begin{equation} \square\delta g_{\mu\nu}' +2R_{\kappa\mu\lambda\nu}\delta{g^{\kappa\lambda}}'=0. \end{equation} For the Schwarzschild metric in particular using this equation results in the Regge-Wheeler equation.

I don't know why you need AdS/CTF correspondence for that? There is a good derivation found here (page 64 to 90). To give a quick overview:

You can calculate the propagation of gravitational waves on a background metric using pertubation theory in first order. You start with a pertubation of the metric tensor given by: \begin{equation} \widetilde{g}_{\mu\nu} =g_{\mu\nu}+\delta g_{\mu\nu} \qquad\text{with}\qquad |\delta g_{\mu\nu}|<1 \end{equation} \begin{equation} \widetilde{g}^{\mu\nu} =g^{\mu\nu}-\delta g^{\mu\nu} \qquad\text{with}\qquad |\delta g_{\mu\nu}|>1. \end{equation} The different signs were explained by me here already. You can now calculate the Christoffel symbols, the Riemann curvature tensor and the Einstein tensor only in first order of the pertubation.

In a vacuum, we get the equation $G_{\mu\nu}[g]=0$ (equivalent to $R_{\mu\nu}[g]=0$) for the background metric and $\delta G_{\mu\nu}=0$ for the gravitational waves out of the field equations. Using a corresponding pertubation tensor (similar to the correspondence between the Ricci and Einstein tensor): \begin{equation} \delta g_{\mu\nu}' =\delta g_{\mu\nu}-\frac{1}{2}(g^{\kappa\lambda}\delta g_{\kappa\lambda})g_{\mu\nu}, \end{equation} \begin{equation} \delta g_{\mu\nu} =\delta g_{\mu\nu}'-\frac{1}{2}(g^{\kappa\lambda}\delta g_{\kappa\lambda}')g_{\mu\nu}. \end{equation} the equation describing the scattering of gravitational waves is given by: \begin{equation} \square\delta g_{\mu\nu}' +2R_{\kappa\mu\lambda\nu}\delta{g^{\kappa\lambda}}'=0. \end{equation} For the Schwarzschild metric in particular using this equation results in the Regge-Wheeler equation.

I don't know why you need AdS/CTF correspondence for that? There is a good derivation found here (page 64 to 90). To give a quick overview:

You can calculate the propagation of gravitational waves on a background metric using pertubation theory in first order. You start with a pertubation of the metric tensor given by: \begin{equation} \widetilde{g}_{\mu\nu} =g_{\mu\nu}+\delta g_{\mu\nu} \qquad\text{with}\qquad |\delta g_{\mu\nu}|<1 \end{equation} \begin{equation} \widetilde{g}^{\mu\nu} =g^{\mu\nu}-\delta g^{\mu\nu} \qquad\text{with}\qquad |\delta g_{\mu\nu}|>1. \end{equation} The different signs were explained by me here already. You can now calculate the Christoffel symbols, the Riemann curvature tensor and the Einstein tensor only in first order of the pertubation.

In a vacuum, we get the equation $G_{\mu\nu}[g]=0$ (equivalent to $R_{\mu\nu}[g]=0$) for the background metric and $\delta G_{\mu\nu}=0$ for the gravitational waves out of the field equations. Using a corresponding pertubation tensor (similar to the correspondence between the Ricci and Einstein tensor): \begin{equation} \delta g_{\mu\nu}' =\delta g_{\mu\nu}-\frac{1}{2}(g^{\kappa\lambda}\delta g_{\kappa\lambda})g_{\mu\nu}, \end{equation} \begin{equation} \delta g_{\mu\nu} =\delta g_{\mu\nu}'-\frac{1}{2}(g^{\kappa\lambda}\delta g_{\kappa\lambda}')g_{\mu\nu}, \end{equation} the equation describing the scattering of gravitational waves is given by: \begin{equation} \square\delta g_{\mu\nu}' +2R_{\kappa\mu\lambda\nu}\delta{g^{\kappa\lambda}}'=0. \end{equation} For the Schwarzschild metric in particular using this equation results in the Regge-Wheeler equation.