Asymptotic Quasinormal Frequencies of Brane-Localized Black Hole

TL;DR

This paper calculates the asymptotic quasinormal frequencies of brane-localized higher-dimensional black holes, revealing how these frequencies depend on the number of extra dimensions and differ from the standard four-dimensional case.

Contribution

It provides the first detailed analysis of how extra dimensions influence the asymptotic quasinormal frequencies of brane-localized black holes, especially for scalar and gravitational perturbations.

Findings

Reproduction of $Re(\,\omega / T_H) = \ln 3$ for $n=0$ in scalar perturbation.

For $n > 4$, $Im(\,\omega / T_H)$ vanishes in scalar perturbation.

$Re(\,\omega / T_H)$ approaches approximately 0.906 as $n \to \infty$.

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 $ \omega$ 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 $Re(\omega / T_H) = \ln 3$ is reproduced when $n = 0$. For $n \neq 0$, however, $Re(\omega / T_H)$ is smaller than $\ln 3$. It is shown also that when $n > 4$, $Im(\omega / T_H)$ vanishes in the scalar perturbation. For the gravitational perturbation it is shown that $Re(\omega / T_H) = \ln 3$ is reproduced when $n = 0$ and $n = 4$. For different $n$, however, $Re(\omega / T_H)$ is smaller than $\ln 3$. When $n = \infty$, for example, $Re(\omega / T_H)$ approaches to $\ln (1 + 2 \cos \sqrt{5} \pi) \approx 0.906$. Unlike the scalar perturbation $Im(\omega / T_H)$ does not vanish regradless of the number of extra dimensions.