Quasinormal modes of tensor perturbation in Kaluza-Klein black hole for Einstein-Gauss-Bonnet gravity

Abstract

In Einstein-Gauss-Bonnet gravity, we study the quasi-normal modes (QNMs) of the tensor perturbation for the so-called Maeda-Dadhich black hole which locally has a topology $\mathcal{M}^n \simeq M^4 \times \mathcal{K}^{n-4}$. Our discussion is based on the tensor perturbation equation derived in~\cite{Cao:2021sty}, where the Kodama-Ishibashi gauge invariant formalism for Einstein gravity theory has been generalized to the Einstein-Gauss-Bonnet gravity theory. With the help of characteristic tensors for the constant curvature space $\mathcal{K}^{n-4}$, we investigate the effect of extra dimensions and obtain the scalar equation in four dimensional spacetime, which is quite different from the Klein-Gordon equation. Using the asymptotic iteration method and the numerical integration method with the Kumaresan-Tufts frequency extraction method, we numerically calculate the QNM frequencies. In our setups, characteristic frequencies depend on six distinct factors. They are the spacetime dimension $n$, the Gauss-Bonnet coupling constant $α$, the black hole mass parameter $μ$, the black hole charge parameter $q$, and two ``quantum numbers" $l$, $γ$. Without loss of generality, the impact of each parameter on the characteristic frequencies is investigated while fixing other five parameters. Interestingly, the dimension of compactification part has no significant impact on the lifetime of QNMs.

Figures (14)

• Figure 1: The AIM numerical results of iteration order from $1$st to $50$th. The parameter choice is as shown in the figure. The horizontal axis stands for the real part of frequencies, while the vertical axis stands for the imaginary part of frequencies. Points of different color are from different iteration orders, as indicated by the legend bar on the right. Since we expect the physical frequencies repeatedly appear in the solution of each iteration order, we may recognize the common position of various different colors to be the location of physical frequencies, while those points which show only one color are considered as numerical artifacts.
• Figure 2: The variance estimate for QNM frequencies with overtone $\hat{\mathcal{n}}=1$. The blue points are the results for the $\hat{\mathcal{n}}=1$ frequencies from the highest 11 iteration orders (the $40$-th order to $50$-th order, here), and the orange point stands for the mean value of them. The light and deep yellow regions indicate $2\sigma$ and $1\sigma$ regions, respectively.
• Figure 3: The AIM calculation result with the parameter indicated in the figure. The physical parameters are unchanged, while the expanding positions is chosen to be $\xi_0=0.5$. The left panel shows the numerical result for different orders, and the right panel is for the mean value and variance of the corresponding $\hat{\mathcal{n}}=1$ QNM frequency.
• Figure 4: These figures show how the variances defined in \ref{['eq: AIM_variance']} of QNM frequencies changes with respect to changing expanding position. The horizontal axis stands for the expanding position, while the vertical axis is for the value of the variance. The three figures are for $\hat{\mathcal{n}}=0$, $\hat{\mathcal{n}}=1$ and $\hat{\mathcal{n}}=2$, respectively. Different color lines stands for different iteration orders.
• Figure 5: These figures show how the mean value of QNM frequencies changes with respect to changing expanding position. The horizontal axis stands for the expanding position, while the vertical axis is for the mean value. The six figures are for $\hat{\mathcal{n}}=0$, $\hat{\mathcal{n}}=1$ and $\hat{\mathcal{n}}=2$, from left to right, respectively. The upper panel is for the real part, while the bottom panel is for the imaginary part. Different color lines stands for different iteration orders.