Scalar gravitational perturbations and quasinormal modes in the five dimensional Schwarzschild black hole

We calculate the quasinormal modes (QNMs) for gravitational perturbations of the Schwarzschild black hole in the five dimensional (5D) spacetime with a continued fraction method. As shown by Kodama and Ishibashi, the gravitational perturbations of higher-dimensional (higher-D) Schwarzschild black holes can be divided into three decoupled classes, namely scalar-gravitational, vector-gravitational, and tensor-gravitational perturbations. In order to examine the QNMs, we make use of Schr\"odinger-type wave equations for determining the dynamics of the gravitational perturbations. We apply the continued fraction method and expand the eigenfunctions around the black hole horizon in terms of Fr\"obenius series. It is found that the resulting recurrence relations become an eight-term relation for the scalar-gravitational perturbations and four-term relations for the vector-gravitational and tensor-gravitational perturbations. For all the types of perturbations, the QNMs associated with $l=2$, $l=3$, and $l=4$ are calculated. Our numerical results are summarized as follows: (i) The three types of gravitational perturbations associated with the same angular quantum number $l$ have a different set of the quasinormal (QN) frequencies; (ii) There is no purely imaginary frequency mode; (iii) The three types of gravitational perturbations have the same asymptotic behavior of the QNMs in the limit of the large imaginary frequencies, which are given by $\omega T_H^{-1}\to\log{3}+ 2\pi i (n+1/2)$ as $n\to\infty$, where $\omega$, $T_H$, and $n$ are the oscillation frequency, the Hawking temperature of the black hole, and the mode number, respectively.