Schwarzschild black hole in the dilatonic domain wall

In the dilatonic domain wall model, we study the Schwarzschild black hole as a solution to the Kaluza-Klein (KK) zero mode effective action which is equivalent to the Brans-Dicke (BD) model with a potential. This can describe the large Randall-Sundrum (RS) black hole whose horizon is to be the intersection of the black cigar with the brane. The black cigar located far from the AdS${}_5$-horizon is known to be stable, but any explicit calculation for stability of the RS black hole at $z=0$ is not yet performed. Here its stability is investigated against the $z$-independent perturbations composed of odd, even parities of graviton ($h_{\mu \nu }$) and BD scalar($h_{44}=2\varphi$). It seems that the RS black hole is classically unstable because it has a potential instability at wavelength with $\lambda > 1/(2k)$\mathrm{.}However,thisisnotallowedinsideanAdS$_5$-boxofthesizewith$1/(2k)$. Thus the RS black hole becomes stable. The RS black hole can be considered as a stable remnant at $z=0$ of the black cigar.