r_c^2
d\phi^2
\hspace{2.0cm} \mbox{(RS1)}     \\   \nonumber
ds^2&=&e^{-2k |y|} \eta_{\mu \nu} dx^{\mu} dx^{\nu} + dy^2
\hspace{2.7cm} \mbox{(RS2)}\end{aligned}$$ where $|\phi| \leq \pi$ and $|y| < \infty$. The parameter $r_c$ is a radius of the compactified fifth dimension. The anology of RS scenario to $AdS$/CFT[@mal98] enables us to explore the finite temperature effect in RS brane-world scenario by extending $AdS_5$ bulk spacetime to Schwarzschild-$AdS_5$[@park01-1; @park01-2].

Inserting the small fluctuation equations $$\begin{aligned}
\label{fluctua}
ds^2&=&\left(e^{-2kr_c |\phi|} \eta_{\mu \nu} + h_{\mu \nu}(x, \phi) \right)
dx^{\mu} dx^{\nu} + r_c^2 d\phi^2
\hspace{1.5cm} \mbox{(RS1)}     \\   \nonumber
ds^2&=&\left(e^{-2k |y|} \eta_{\mu \nu} + h_{\mu \nu}(y) \right)
dx^{\mu} dx^{\nu} + dy^2
\hspace{2.7cm} \mbox{(RS2)}\end{aligned}$$ to $5d$ Einstein equation one can derive a gravitational fluctuation equation $$\begin{aligned}
\label{grafluc}
& &\hat{H}_{RS} \hat{\psi}(z) = \frac{m^2}{2} \hat{\psi}(z)  \\  \nonumber
& &\hat{H}_{RS} = -\frac{1}{2} \partial_z^2 + V_i(z)\end{aligned}$$ where $i=1.2$ represents $i^{th}$ RS scenario. For each RS scenario the potential becomes $$\begin{aligned}
\label{potential}
V_1(z)&=&\frac{15 k^2}{8(k|z| + 1)^2} - \frac{3}{2} k
\left[ \delta(z) - \delta(z - z_0) \right]   \\   \nonumber
V_2(z)&=&\frac{15 k^2}{8(k|z| + 1)^2} - \frac{3}{2} k \delta(z)\end{aligned}$$ where $z_0 = (e^{k r_c \pi} - 1) / k$. The function $\hat{\psi}(z)$ is related to the linearized gravitational field $h$ as follows $$\begin{aligned}
\label{relation1}
h(x, \phi)&=& e^{-\frac{k}{2} r_c |\phi|} \hat{\psi} e^{ipx}
