Slowly damped quasinormal modes of the massive Dirac field in $d$-dimensional Tangherlini spacetime

We consider quasinormal modes of the massive Dirac field in the background of a Schwarzschild-Tangherlini black hole. Different dimensions of the spacetime are considered, from $d=4$ to $d=9$. The quasinormal modes are calculated using two independent methods: WKB and continued fraction. We obtain the spectrum of quasinormal modes for different values of the overtone number and angular quantum number. An analytical approximation of the spectrum valid in the case of large values of the angular quantum number and mass is calculated. Although we do not find unstable modes in the spectrum, we show that for large values of the mass, the quasinormal modes can become very slowly damped, giving rise to quasistationary perturbations.

Figure 1

The fundamental quasinormal modes in the complex $\mathrm{\Omega }$-plane for $l=10$. The crosses and dots are the results from the 3rd order WKB method with $\mathrm{sgn}(\kappa )=+1$ and $\mathrm{sgn}(\kappa )=-1$, respectively. The difference in mass value between two adjacent points is $\mathrm{\Delta }\eta =0.5$. The solid lines are the analytical results from the large $\kappa$ approximation using the first order WKB approximation. From left to right in different colors $d=4$, 5, 6, 7, 8, 9. Marked with grey triangles are the quasinormal modes for $\eta =0$. With increasing $\eta$, the value of the $|\Im (\mathrm{\Omega })|$ becomes smaller and the real part becomes larger. Around the red circles, for $d=4$, 5 the imaginary part becomes very close to zero, and the numerical methods break down.

Figure 2

Similar figure to the previous one, showing the imaginary part of $\mathrm{\Omega }$ over the mass parameter $\eta$ for the fundamental $l=10$ mode. We normalize with respect to the imaginary part of the frequency for $\eta =0$.

Figure 3

Real part of the fundamental modes for $d=4$, 5 and $l=10$ ($\mathrm{sgn}(\kappa )=+1$), as a function of the mass $\eta$. The dotted green line marks $\Re (\mathrm{\Omega })=\eta$. The dash-dotted curves are the analytical approximation in the large $\kappa$ limit using the first order WKB approximation. The circle points are the results from the 3rd order WKB method and the solid orange curves are the results from the continued fraction method. The red circles mark the region where the WKB, continued fraction method and analytical approximation break down.

Figure 4

Real part of the point $x_0$ for which $V_{\mathrm{eff}}^{\prime }(x_0)=0$ over the mass $\eta$ for $l=10$ ($\mathrm{sgn}(\kappa )=+1$). The black and blue solid curves are the large $\kappa$ approximation for $d=4$, 5 respectively. The circle points were obtained using the full effective potential. The vertical solid red line marks the values of the mass for which $x_0$ diverges in five dimensions at ${\eta }_{00}^{d=5}=\kappa$. The orange vertical line marks the value of ${\eta }_{00}^{d=4}=\kappa /\sqrt{3}$.

Figure 5

Imaginary part of $\mathrm{\Omega }$ over the real part of $\mathrm{\Omega }$ for different overtone numbers $n$ for the five dimensional case with $l=10$ and $\mathrm{sgn}(\kappa )=+1$. From bottom to top $n=0$, 1, 2 and 3. The circles are the numerical results using the third order WKB approximation using the full effective potential. The solid lines are the analytical values in the large $\kappa$ approximation of the effective potential also using the third order WKB approximation. For the full potential the numerical values are displayed up to the point the WKB and continued fraction method begin to diverge. Marked with grey triangles are the points with $\eta =0$.

Figure 6

Similar figure to the previous one, here presenting the imaginary part of $\mathrm{\Omega }$ over the mass $\eta$ for different overtone numbers $n$ for the five dimensional case with $l=10$ and $\mathrm{sgn}(\kappa )=+1$.

Figure 7

Fundamental modes in the complex $\mathrm{\Omega }$-plane for $l=0$, using the continued fraction method. From the left to the right we have $d=4$, 5, 6, 7, 8 and 9. Shown in continuous lines are the values for $\mathrm{sgn}(\kappa )=+1$ and in dashed lines the values for $\mathrm{sgn}(\kappa )=-1$. Marked with black triangles are the quasinormal modes for $\eta =0$. We only show modes with a relative error of 0.5%, except in $d=4$ and 5 close to the real axis, where the numerical method breaks down, and we relax the condition to a relative error of 2%.

Figure 8

A similar figure to the previous one, showing the real part of the fundamental mode as a function of the mass for $l=0$, and different values of the spacetime dimension $d$. In grey we include the line $\Re (\mathrm{\Omega })=\eta$.

Figure 9

First excited quasinormal modes in the complex $\mathrm{\Omega }$-plane for $l=0$, using the continued fraction method. From the bottom left to the upper right $d=4$, 5, 6, 7, 8 and 9. Shown in continuous lines are the values for $\mathrm{sgn}(\kappa )=+1$ and in dashed lines the values for $\mathrm{sgn}(\kappa )=-1$. Marked with black triangles are the quasinormal modes for $\eta =0$. We show modes with relative error smaller than 0.5%.

Figure 10

Similar figure to the previous one, showing the real part of the quasinormal modes for the first excited modes for $l=0$. In grey we mark the line $\Re (\mathrm{\Omega })=\eta$.