Asymptotic Quasinormal Frequencies of Brane-Localized Black Hole

Abstract

The asymptotic quasinormal frequencies of the brane-localized $(4+n)$-dimensional black hole are computed. Since the induced metric on the brane is not an exact vacuum solution of the Einstein equation defined on the brane, the real parts of the quasinormal frequencies $ ω$ do not approach to the well-known value $T_H \ln 3$ but approach to $T_H \ln k_n$, where $k_n$ is a number dependent on the extra dimensions. For the scalar perturbation $Re(ω/ T_H) = \ln 3$ is reproduced when $n = 0$. For $n \neq 0$, however, $Re(ω/ T_H)$ is smaller than $\ln 3$. It is shown also that when $n > 4$, $Im(ω/ T_H)$ vanishes in the scalar perturbation. For the gravitational perturbation it is shown that $Re(ω/ T_H) = \ln 3$ is reproduced when $n = 0$ and $n = 4$. For different $n$, however, $Re(ω/ T_H)$ is smaller than $\ln 3$. When $n = \infty$, for example, $Re(ω/ T_H)$ approaches to $\ln (1 + 2 \cos \sqrt{5} π) \approx 0.906$. Unlike the scalar perturbation $Im(ω/ T_H)$ does not vanish regradless of the number of extra dimensions.