Quasinormal modes of nearly extremal Kerr spacetimes: Spectrum bifurcation and power-law ringdown

We provide an in-depth investigation of quasinormal-mode oscillations of Kerr black holes with nearly extremal angular momenta. We first discuss in greater detail the two distinct types of quasinormal-mode frequencies presented in a recent paper [H. Yang et al., Phys. Rev. D 87, 041502 (2013)]. One set of modes that we call “zero-damping modes” has vanishing imaginary part in the extremal limit and exists for all corotating perturbations (i.e. modes with azimuthal index $m\ge 0$). The other set (the “damped modes”) retains a finite decay rate even for extremal Kerr black holes and exists only for a subset of corotating modes. As the angular momentum approaches its extremal value, the frequency spectrum bifurcates into these two distinct branches when both types of modes are present. We discuss the physical reason for the mode branching by developing and using a bound-state formulation for the perturbations of generic Kerr black holes. We also numerically explore the specific case of the fundamental $l=2$ modes, which have the greatest astrophysical interest. Using the results of these investigations, we compute the quasinormal-mode response of a nearly extremal Kerr black hole to perturbations. We show that many superimposed overtones result in a slow power-law decay of the quasinormal ringing at early times, which later gives way to exponential decay. This exceptional early-time power-law decay implies that the ringdown phase is long lived for black holes with large angular momentum, which could provide a promising strong source for gravitational-wave detectors.

Figure 1

WKB results for $r_0$, ${\Omega }_R-\mu /2$, and ${\Omega }_I$, for various values of $\mu$ and for increasing angular momenta. The angular momenta are $a=0.9$ (blue, dashed-dotted lines), $a=0.999$ (red, dotted lines), and $a=0.99999$ (black, solid lines). Left: The radius $r_0$, which coincides with the horizon for ${\mu }_c<\mu <1$ as $a\to 1$. Middle: The scaled frequency minus the scaled angular frequency of the horizon, ${\Omega }_R-\mu /2$. Right: The decay rate ${\Omega }_I$, which vanishes for ${\mu }_c<\mu <1$ as $a\to 1$.

Figure 2

Plot of the potential term Eq. (3.1) for different $\mu$. Here $\mu =0.4$, 0.5, 0.6, 0.7, 0.8, and 0.9 correspond to black solid, red dashed, red dotted, blue dotted, blue dashed, and magenta solid curves, respectively. The transition happens between 0.7 to 0.8. Reproduced from 16.

Figure 3

Comparison between numerical (circles and squares) and analytical (solid lines) calculations for the $a\to 1$ behavior of the $(l,m)=(2,2)$ mode. Here red square (blue circle) stands for the real (imaginary) part of $\eta$. The inset shows that, up to $ϵ=0.01$, the real and imaginary parts of $\eta$ are still bounded by $\pm 0.04$.

Figure 4

Magnitude of $\eta$ as a function of $\delta$. The red dashed line is for $n=0$ given by the lower bound in Eq. (3.24). The two gray lines are for $n=1$ and $n=5$. The solid red line corresponds to $n\to \infty$, and is given by the upper bound of Eq. (3.25).

Figure 5

ZDM wave functions in the bound-state picture for $(l,m,s)=(2,1,0)$. Black solid, red dashed, blue dotted, blue dashed, and magenta solid lines correspond to $n=0$, 1, 2, 3, 4 respectively. The wave functions approach $0$ for $r\to r_{+}$ or $\sqrt{ϵ}{r}_{*}\to 0$, and stay negligibly small for $r-1\gg \sqrt{ϵ}$. Therefore the wave function is bounded in the near-horizon regime. In addition, as we increase the overtone $n$, the wave functions monotonically move away from the horizon. This is a general feature for all ZDMs.

Figure 6

Phase diagram for the separation between the single- and double-branch regime for NEK BHs. Large purple dots ($s=-2$) and gold crosses ($s=0$) correspond to $(l,m)$ pairs with only ZDMs, while smaller blue dots correspond to $(l,m)$ pairs with both ZDMs and DMs. The green line is the phase boundary computed using the eikonal approximation. Reproduced from 16.

Figure 7

Contours are constant values of the logarithm of the continued fraction for the case $s=-2$, $l=2$, $m\le 2$. Darker shading indicates values close to zero. The contours cluster around the QNM values in darker regions and around poles in the fraction for lighter regions. The $+$’s correspond to the analytic ZDM prediction, Eq. (3.27).

Figure 8

Plots of QNM frequencies $\omega$ for the case $s=-2$, $(l,m)=(2,1)$ values as found using Leaver’s method (with inversion). The first six overtones (at low angular momenta) are shown, which become two DMs and the first four ZDMs as we increase $a$ from 0.9 to 0.99999, using logarithmically decreasing spacing. The decay rate ${\omega }_I$ of the two DMs changes relatively little as $a\to 1$, while the ZDMs move towards their extremal limit $\omega \to m/2$.

Figure 9

Plots of QNM frequencies $\omega$ for the case $s=0$, $(l,m)=(10,7)$ values as found using Leaver’s method (with inversion). The first seven overtones (at low angular momenta) are shown, which become three DMs and the first four ZDMs as we increase $a$ from 0.99 to 0.99999, using logarithmically decreasing spacing.

Figure 10

Estimates for the (scaled) critical values $L^2{ϵ}_c$ for which $\mathcal{I}m({\omega }_{\mathrm{ZDM}})=\mathcal{I}m({\omega }_{\mathrm{DM}})$, when the lowest-order ZDM becomes the least-damped QNM. All estimates for modes with $2\le l\le 15$ and $0\le m<(l+1/2){\mu }_c$ are plotted, with lines linking estimates with the same index $l$. The scaled values cluster onto a curve which depends only on $\mu$.

Figure 11

The scalar QNM response of a hole with $ϵ={10}^{-4}$, as described by Eq. (5.11) for a scalar $l=2$, $m=2$ perturbation, with a radial $\delta$-function distribution. The initial amplitude is normalized to unity. The blue curve plots the logarithm of $\Re [\Phi ](T)$, and its envelope is shown by a black dotted line. Also plotted is a curve $1/T$ (purple dot-dashed line) and the decay envelope for the $n=0$ ZDM (red dashed line).

Figure 12

Time domain signal for the $l=2$, $m=2$ mode measured at future null infinity for various angular momenta. Early-time polynomial ringdown is observed longer for smaller $ϵ\equiv 1-a$ in accordance with Eq. (5.1) (compare also Fig. 11).

Figure 13

Comparison of analytic formula and numerical calculation after the peak. The dots are the numerical maxima in the previous figure. The solid lines are computed from the analytic formula Eq. (5.19).

Figure 14

Plots of QNM frequencies $\omega$ for the case $s=0$, $(l,m)=(2,1)$ values as found using Leaver’s method (with inversion). The first seven overtones (at low angular momenta) are shown, which become for three DMs and the first four ZDMs as we increase $a$ from 0.950 to 0.99999 using logarithmically decreasing spacing. Left panel: All seven modes visualized. The first two ZDMs momentarily join the third DM, as discussed in the text. Right panel: The third DM executes spiraling motion, which we have not attempted to resolve; the first two sharp jumps occur because of interaction with the first two ZDMs.

Figure 15

The exchange of QNM identity for the $s=0$, (2, 1) mode illustrated by continued-fraction value as plotted in Fig. 14. Top panel: At $a=0.973$, the third and fourth overtones approach each other. Middle panel: At $a=0.975$ the fourth overtone lies just above the third. Bottom panel: As we raise the angular momentum to $a=0.977$, we see the third overtone begins to move toward the real axis, becoming the first ZDM, while the fourth becomes the third DM.