Gravitational perturbations of the Kerr geometry: High-accuracy study

We present results from a new code for computing gravitational perturbations of the Kerr geometry. This new code carefully maintains high precision to allow us to obtain high-accuracy solutions for the gravitational quasinormal modes of the Kerr space-time. Part of this new code is an implementation of a spectral method for solving the angular Teukolsky equation that, to our knowledge, has not been used before for determining quasinormal modes. We focus our attention on two main areas. First, we explore the behavior of these quasinormal modes in the extreme limit of Kerr, where the frequency of certain modes approaches accumulation points on the real axis. We compare our results with recent analytic predictions of the behavior of these modes near the accumulation points and find good agreement. Second, we explore the behavior of solutions of modes that approach the special frequency $M\omega =-2i$ in the Schwarzschild limit. Our high-accuracy methods allow us to more closely approach the Schwarzschild limit than was possible with previous numerical studies. Unlike previous work, we find excellent agreement with analytic predictions of the behavior near this special frequency. We include a detailed description of our methods, and make use of the theory of confluent Heun differential equations throughout. In particular, we make use of confluent Heun polynomials to help shed some light on the controversy of the existence, or not, of quasinormal and total-transmission modes at certain special frequencies in the Schwarzschild limit.

Figure 1

Magnitude of the spectral coefficients $C_{{\ell }^{\prime }\ell m}(c)$ along Kerr QNM sequences. Each line plots $\log |C_{{\ell }^{\prime }\ell m}(c)|$, for a particular ${\ell }^{\prime }$ and with $c=\overline{a}\overline{\omega }$ along a particular QNM sequence with $0\le \overline{a}<1$. The upper plot shows the coefficient of the fundamental mode for $\ell =2$, $m=2$ (see Fig. 3). The lower plot shows the coefficients of the fundamental mode for $\ell =6$, $m=2$ (see Fig. 9).

Figure 2

Kerr QNM mode sequences for $m=2$. The complex frequency $\overline{\omega }$ is plotted for the cases $\ell =2\to 12$ and $n=0\to 7$. Note that the imaginary axis is inverted. Each sequence covers the range $0\le \overline{a}<1$, with markers on each sequence denoting a change in $\overline{a}$ of 0.05. The $\overline{a}=0$ element of each set of mode sequences with the same $\ell$ are connected by a dashed line. $\ell$ increases monotonically as we move to the right in the plot. The overtone index $n$ for each set of modes with the same $\ell$ increases monotonically as we move up along each dashed line.

Figure 3

Kerr QNM mode sequences for $\ell =2$ and $m=2$. The left plot displays the complex frequency $\overline{\omega }$ and the right plot displays the separation constant ${}_{-2}{A}_{22}(\overline{a}{\overline{\omega }}_{22n})$ for corresponding mode sequences. The left plot is as described in the caption for Fig. 2. For the right plot, each sequence begins with $\overline{a}=0$ on the real axis at $\ell (\ell +1)-2$. The overtones all radiate from this point with $n$ increasing in a clockwise direction. $n=0\to 7$ are displayed for both plots.

Figure 4

Close-up of $\overline{\omega }$ for the {2, 2, 5} sequence showing the unusual termination of this sequence. Instead of approaching the accumulation point at $m/2$, this sequence makes a sharp loop and then terminates with finite damping. See Fig. 3 for context.

Figure 5

Kerr QNM mode sequences for $\ell =3$ and $m=2$. Plots are as described in Fig. 3.

Figure 6

Kerr QNM mode sequences for $\ell =4$ and $m=2$. Plots are as described in Fig. 3.

Figure 7

Close-up of $\overline{\omega }$ for the {4, 2, 6} sequence showing the unusual termination of this sequence. See Fig. 6 for context.

Figure 8

Kerr QNM mode sequences for $\ell =5$ and $m=2$. Plots are as described in Fig. 3.

Figure 9

Kerr QNM mode sequences for $\ell =6$ and $m=2$. Plots are as described in Fig. 3.

Figure 10

Kerr QNM mode sequences for $m=1$. See Fig. 2 for a full description.

Figure 11

Kerr QNM mode sequences for $\ell =2$ and $m=1$. Plots are as described in Fig. 3.

Figure 12

Kerr QNM mode sequences for $\ell =3$ and $m=1$. Plots are as described in Fig. 3.

Figure 13

Close-up of $\overline{\omega }$ for the {3, 1, 6} sequence showing the unusual termination of this sequence. See Fig. 12 for context.

Figure 14

Kerr QNM mode sequences for $m=0$. See Fig. 2 for a full description.

Figure 15

Kerr QNM mode sequences for $\ell =2$ and $m=0$. Plots are as described in Fig. 3.

Figure 16

Close-up of $\overline{\omega }$ for the $\{2,0,5–7\}$ sequence showing the unusual termination of these sequences. See Fig. 15 for context.

Figure 17

Kerr QNM mode sequences for $m=-1$. See Fig. 2 for a full description.

Figure 18

Kerr QNM mode sequences for $m=-2$. See Fig. 2 for a full description.

Figure 19

Near extremal behavior of $\overline{\omega }$ for $\ell =5$, $m=4$. Symbols in the left plot represent data values used to fit the coefficients in Eq. (63a). The lines represent the resulting fit function. The right plot corresponds to Eq. (63b). In both plots, the dashed (red) line represents the fit function omitting the next to leading order behavior in $\epsilon$.

Figure 20

Near extremal behavior of ${}_{-2}{A}_{\ell m}(\overline{a}\overline{\omega })$ for $\ell =5$, $m=4$. Symbols in the left plot represent data values used to fit the coefficients in Eq. (63c). The solid lines represent the resulting fit function. The right plot corresponds to Eq. (63d). In both plots, the dashed (red) line represents the fit function omitting the next to leading order behavior in $\epsilon$.

Figure 21

Near extremal behavior of $\overline{\omega }$ for $\ell =2$, $m=1$. Symbols in the left plot represent data values used to fit the coefficients in Eq. (64a). The lines represent the resulting fit function. The right plot corresponds to Eq. (64b). In the right plot, the nearly imperceptible dashed (red) line represents the fit function omitting the next to leading order behavior in $\epsilon$.

Figure 22

Near extremal behavior of ${}_{-2}{A}_{\ell m}(\overline{a}\overline{\omega })$ for $\ell =2$, $m=1$. Symbols in the left plot represent data values used to fit the coefficients in Eq. (64c). The lines represent the resulting fit function. The right plot corresponds to Eq. (64d). In the right plot, the nearly imperceptible dashed (red) line represents the fit function omitting the next to leading order behavior in $\epsilon$.

Figure 23

Behavior of $\overline{\omega }$ for all $n=8$ overtones of $\ell =2$. The left plot shows the full extent of all sequences with $-2\le m\le 2$ and clearly shows that the $m=1$ and $m=2$ modes are both multiplets. The multiplets are distinguished by a subscript on the overtone index. The right plot shows a close-up of the behavior near the NIA. Note that the $m=1$, 2 modes for the $n=8_0$ multiplet approach the NIA near $\overline{\omega }=-1.964$ while the modes for the $n=8_1$ multiplet approach the NIA near $\overline{\omega }=-2.042$.

Figure 24

Kerr algebraically special TTM mode sequences for $\ell =2$, $m=0–2$ and for $\ell =3$, $m=0–3$. Note that the imaginary axis is inverted and shifted to the left so the $m=0$ behavior can be more easily seen. Each sequence covers a range $0\le \overline{a}\le 1$, with markers on each sequence denoting a change in $\overline{a}$ of 0.05.