Figure 1

Numerically computed admissible range of parameters for the subextremal Kerr–de Sitter black hole (light shaded) [30] and the range to which our results apply (dark shaded). QNMs are defined and discrete for parameters below the dashed line, $(1-\alpha {)}^3=9\Lambda M^2$; see 8, Sec. 3.2.Reuse & Permissions

Figure 2

Visualization of the trapped set for different values of $a$, with $\Lambda =0$. The figures show the four-dimensional set $K\cap \{{\xi }_t=1\}$ ($K$ is the five-dimensional trapped set) projected to the coordinates $(x,y,z)=({\xi }_{\phi },\theta ,{\xi }_{\theta })$. For $a=0$ this corresponds to the visualization of the phase space of the 2-sphere: the sphere is parametrized by the coordinates $0\le \theta \le \pi$, $0\le \phi <2\pi$, ($\sin \theta \cos \phi$, $\sin \theta \sin \phi$, $\cos \theta$). The conjugate coordinates are denoted ${\xi }_{\theta }$ and ${\xi }_{\phi }$ and the restriction to ${\xi }_t=1$ means that ${\xi }_{\theta }^2+{sin}^{-2}\theta {\xi }_{\phi }^2=27M^2$. The vertical singular interval in front corresponds to $\theta =0$, with the symmetrical interval in the back corresponding to $\theta =\pi$: the coordinates $(\theta ,\phi )$ on the sphere are singular at that point. The structure of $K$ becomes more interesting when $a>0$ as shown in the three examples. The additional coordinate, not shown in the figures, $r$ is a function of ${\xi }_{\phi }$ and ${\xi }_t$ only. When $a=0$, we have $r=3M$, but $r$ gets larger to the left (${\xi }_{\phi }>0$) and smaller to the right (${\xi }_{\phi }<0$) when $a>0$. When $a=1$ we see the flattening in the $(\theta ,{\xi }_{\theta })$ plane at extremal values of ${\xi }_{\phi }$: the trapped set touches the event horizon $r=1$ which results in lack of decay, and some QNMs have null imaginary parts [22, 23]. Dynamically and invariantly this corresponds to the vanishing expansion rates—see Fig. 4.Reuse & Permissions

Figure 3

The dependence of ${\nu }_{\max }$ and ${\nu }_{\min }$ on the parameters $M$ and $a$ in the case of $\Lambda =0$. The dashed line indicates the range of validity of the pinching condition needed for the Weyl law (12).Reuse & Permissions

Figure 4

The pointwise expansion rates $\nu$ on Liouville tori ${\xi }_{\phi }/(M{\xi }_t)=\mathrm{const}$, for $\theta =\pi /2$. When $a$ approaches 1, $\nu =0$ for some values of ${\xi }_{\phi }$ which shows that there is no gap and QNMs can be arbitrarily close the real axis [22, 23].Reuse & Permissions

Figure 5

A log-log plot of relative errors $\frac{|\min |\mathrm{Im}{\omega }_k(\ell )|-{\nu }_{\min }/2|}{\min |\mathrm{Im}{\omega }_k(\ell )|}$ and $\frac{|\max |\mathrm{Im}{\omega }_k(\ell )|-{\nu }_{\max }/2|}{\max |\mathrm{Im}{\omega }_k(\ell )|}$ where ${\omega }_k(\ell )$ are the numerically computed resonances in the first band corresponding to the angular momentum $\ell$ [9] and ${\nu }_{\min }$, ${\nu }_{\max }$ are minimal and maximal expansion rates. The agreement is remarkable when $\ell$ increases.Reuse & Permissions

Figure 6

A schematic comparison between QNMs in a completely integrable case and in the general $r$-normally hyperbolic case. The former lie on a fuzzy lattice and are well approximated by WKB construction based on quantization conditions [20, 21, 22, 23]. When trapping is $r$-normally hyperbolic and the pinching condition ${\nu }_{\max }<2{\nu }_{\min }$ holds, quasinormal modes are still localized to a strip with dynamically determined bounds, and their statistics are given by the Weyl law (12).Reuse & Permissions

Figure 7

Numerically computed constant in (12), $\mathrm{vol}(K\cap \{{\xi }_t^2\le 1\}\cap \{t=0\})/4{\pi }^2$, for $\Lambda =0$, 0.01, 0.02, 0.03 and $M=1$. The vertical line shows the value of $a$ at which the pinching condition (10) fails for $\Lambda =0$. For smaller values of $a$ the Weyl law (12) holds. We note that as $a$ increases the QNMs split (see Fig. 6) and hence we do expect the counting function to decrease, in agreement with the behavior of the volume. From the volume and the gap giving the decay rate one can read off $a$ and $M$.Reuse & Permissions