Modes of the Kerr geometry with purely imaginary frequencies

In this paper, we examine the behavior of modes of the Kerr geometry when the mode’s frequency is purely imaginary. We demonstrate that quasinormal modes must be polynomial in nature if their frequency is purely imaginary, and present a method for computing such modes. The nature of these modes, however, is not always easy to determine. Some of the polynomial modes we compute are quasinormal modes. However, some are simultaneously quasinormal modes and total transmission modes, while others fail to satisfy the requisite boundary conditions for either. This analysis is, in part, an extension of the results known for Schwarzschild black holes, but clarifies misconceptions for the behavior of modes when the black hole has angular momentum. We also show that the algebraically special modes of Kerr with $m=0$ have an additional branch of solutions not seen before in the literature. All of these results are in precise agreement with new numerical solutions for sequences of gravitational quasinormal modes of Kerr spacetime. However, we show that some prior numerical and analytic results concerning the existence of quasinormal modes of Kerr with purely imaginary frequencies were incorrect.

Figure 1

Kerr QNM mode sequences for $m=-4$. The complex frequency $\overline{\omega }$ is plotted for the cases $\ell =4$ and $0\le n\le 31$. 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 overtone is connected by a dashed line. The overtone index $n$ increases monotonically as we move up the dashed line.

Figure 2

Kerr QNM mode sequences for $m=4$. See Fig. 1 for a full description. Note that as $\overline{a}\to 1$ the sequences approach an accumulation point at $\overline{\omega }=m/2$.

Figure 3

Kerr QNM mode sequences for $m=-3$. See Fig. 1 for a full description. In this case, we plot the $\ell =3$ and $\ell =4$ sequences. Sequences with lower $\ell$ are generally to the left of sequences with higher $\ell$.

Figure 4

Kerr QNM mode sequences for $m=3$. See Fig. 1 for a full description. In this case, we plot the $\ell =3$ and $\ell =4$ sequences. Sequences with lower $\ell$ are generally to the left of sequences with higher $\ell$. Note that as $\overline{a}\to 1$ the sequences approach an accumulation point at $\overline{\omega }=m/2$.

Figure 5

Kerr QNM mode sequences for $m=-2$. See Fig. 1 for a full description. In this case, we plot the $\ell =2$, $\ell =3$ and $\ell =4$ sequences. Note that several of the $\ell =2$ sequences terminate at and reemerge from the NIA near $\overline{\omega }\sim 4$, and several of the $\ell =3$ sequences do the same near $\overline{\omega }\sim 7$. This behavior is examined in more detail in Figs. 11 and 12.

Figure 6

Kerr QNM mode sequences for $m=2$. See Fig. 1 for a full description. In this case, we plot the $\ell =2$, $\ell =3$ and $\ell =4$ sequences. Note that as $\overline{a}\to 1$ many of the sequences approach an accumulation point at $\overline{\omega }=m/2$.

Figure 7

Kerr QNM mode sequences for $m=-1$. See Fig. 1 for a full description. In this case, we plot the $\ell =2$, $\ell =3$ and $\ell =4$ sequences.

Figure 8

Kerr QNM mode sequences for $m=1$. See Fig. 1 for a full description. In this case, we plot the $\ell =2$, $\ell =3$ and $\ell =4$ sequences. Note that as $\overline{a}\to 1$ many of the sequences approach an accumulation point at $\overline{\omega }=m/2$.

Figure 9

Overview of Kerr QNM mode sequences for $m=0$. See Fig. 1 for a full description. In this case, we plot the $\ell =2$, $\ell =3$ and $\ell =4$ sequences. The complex behavior near the NIA is examined in more detail in Figs. 13, 14, 15, 16, 17, 18.

Figure 10

Detailed view near the NIA of Kerr QNM mode sequences for $\ell =2$ and $n=8$. See Fig. 1 for additional description. Each sequence is labeled by it values for $\{\ell ,m,n\}$. Note that the $m=0$, 1, and 2 sequences are overtone multiplets as described in the text. We show in Sec. 4 that the four sequences with $m=1$ and 2 must terminate without a mode on the NIA. The $\{2,0,8_1\}$ emerges from the NIA as simultaneously a QNM and a ${\mathrm{TTM}}_L$ with a nonvanishing value of $\overline{a}$. The remaining three sequences branch out from $\overline{\omega }=-2i$ which is also simultaneously a QNM and a ${\mathrm{TTM}}_L$ with $\overline{a}=0$.

Figure 11

Detailed view near the NIA of Kerr QNM mode sequences for $\ell =2$, $m=-2$ and $11\le n\le 19$. See Fig. 1 for additional description. Each sequence labeled by $\{\ell ,m,n\}$ is part of an overtone multiplet. Sequences with a $n_0$ overtone terminate at the NIA, while those with a $n_1$ overtone reemerge from the NIA. We show in Sec. 4 that none of these sequences have a mode precisely on the NIA.

Figure 12

Detailed view near the NIA of Kerr QNM mode sequences for $\ell =3$, $m=-2$ and $24\le n\le 31$. See Fig. 1 for additional description. Each sequence labeled by $\{\ell ,m,n\}$ is part of an overtone multiplet. Sequences with a $n_0$ overtone terminate at the NIA, while those with a $n_1$ overtone reemerge from the NIA. We show in Sec. 4 that none of these sequences have a mode precisely on the NIA.

Figure 13

Detailed view near the NIA of Kerr QNM mode sequences for $m=0$ and $8\le n\le 15$. See Fig. 1 for additional description. Sequences for $\ell =2$, 3, and 4 are shown. All of the $\ell =2$ sequences are overtone multiplets. The $\ell =2$ sequences also show looping behavior where many of the loops have numerous points of tangency with the NIA.

Figure 14

Detailed view near the NIA of Kerr QNM mode sequences for $m=0$ and $16\le n\le 23$. See Fig. 1 for additional description. Sequences for $\ell =2$, 3, and 4 are shown. All of the $\ell =2$ sequences are overtone multiplets. Many of the $\ell =2$ and 3 sequences show looping behavior where many of the loops have numerous points of tangency with the NIA.

Figure 15

Detailed view near the NIA of Kerr QNM mode sequences for $m=0$ and $24\le n\le 31$. See Fig. 1 for additional description. Sequences for $\ell =2$, 3, and 4 are shown. Only the $n\le 26$, $\ell =2$ sequences are overtone multiplets. Many of the sequences show looping behavior where many of the loops have numerous points of tangency with the NIA.

Figure 16

Examples of $\ell =2$, $m=0$ modes that appear to touch the NIA. Shown are the $n=9$, 14, 20, and 26 sequences. Each has a sequence segment that begins at the Schwarzschild limit and appears to terminate on the NIA. Each sequence reemerges from the NIA and then repeatedly loops back with many points of tangency with the NIA. The lower two panels include insets that show an enlarged view of the second segment.

Figure 17

Examples of $\ell =3$, $m=0$ modes that appear to touch the NIA. Shown are the $n=18$, 23, 27, and 31 sequences. Each has a sequence beginning at the Schwarzschild limit and then beginning to exhibit loops. Many of the loops have points of tangency with the NIA. Each panel includes an inset that shows an enlarged view of the sequence near the NIA.

Figure 18

Examples of $\ell =4$, $m=0$ modes that appear to touch the NIA. Shown are the $n=25$, 27, 29, and 31 sequences. Each has a sequence begins at the Schwarzschild limit and then begin to exhibit loops. Many of the loops have points of tangency with the NIA. Each panel includes an inset that shows an enlarged view of the sequence near the NIA.

Figure 19

Polynomial modes with $\overline{\omega }={\overline{\omega }}_{+}$. The dashed gray lines are the possible values of ${\overline{\omega }}_{+}(N_{+},\overline{a})$ for $1\le N_{+}\le 16$. The marker denotes the discrete points where the ${\mathrm{\Delta }}_{q+1}=0$ condition is satisfied.

Figure 20

Polynomial modes with $\overline{\omega }={\overline{\omega }}_{+}$. The upper-left panel shows results for $\ell =2$, the upper-right panel shows $\ell =3$, and the lower-left $\ell =4$. In each panel, the black dots are a subset of solutions that are shown in Sec. 4d to be polynomial solutions that are simultaneously QNMs and ${\mathrm{TTM}}_L\mathrm{s}$. The gray dots are a subset of solutions that are shown to be neither QNM nor TTM.

Figure 21

Polynomial modes with $\overline{\omega }={\overline{\omega }}_{-}$. The upper-left panel shows results for $\ell =2$, the upper-right panel shows $\ell =3$, and the lower-left $\ell =4$. As shown in Sec. 4d, all of these polynomial solutions are QNMs.

Figure 22

Comparison of $\ell =2$ polynomial modes with QNMs interpolated to the NIA. The open circles represent points where QNM sequences become tangent to the NIA. The points of tangency are determined by interpolation. The small solid dots are polynomial modes obeying $\overline{\omega }={\overline{\omega }}_{+}$. Any mismatch in the centering of dots and circles is a graphing artifact. The maximum deviation has an absolute error of $\mathcal{O}({10}^{-7})$. As shown in Sec. 4d, none of these polynomial solutions are QNMs or TTMs.