Branching of quasinormal modes for nearly extremal Kerr black holes

We show that nearly extremal Kerr black holes have two distinct sets of quasinormal modes, which we call zero-damping modes (ZDMs) and damped modes (DMs). The ZDMs exist for all harmonic indices $l$ and $m\ge 0$, and their frequencies cluster onto the real axis in the extremal limit. The DMs have nonzero damping for all black hole spins; they exist for all counterrotating modes ($m<0$) and for corotating modes with $0\le \mu \lesssim {\mu }_c=0.74$ (in the eikonal limit), where $\mu \equiv m/(l+1/2)$. When the two families coexist, ZDMs and DMs merge to form a single set of quasinormal modes as the black hole spin decreases. Using the effective potential for perturbations of the Kerr spacetime, we give intuitive explanations for the absence of DMs in certain areas of the spectrum and for the branching of the spectrum into ZDMs and DMs at large spins.

Figure 1

Phase diagram for the separation between the single- and double-branch regime for NEK BHs. Large purple dots and gold crosses correspond to $(l,m)$ pairs with only ZDMs for perturbations with spin $-2$ and $0$, respectively. 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.

Figure 2

Plot of the potential term for $a=1$, Eq. (7), for $\mu =0.4$, 0.5, 0.6, 0.7, 0.8, and 0.9 (black-solid, red-dashed, red-dotted, blue-dotted, blue-dashed, and magenta-solid curves, respectively). The transition from single-branch to double-branch happens between $\mu =0.7$ and $\mu =0.8$.

Figure 3

QNM frequencies with $l=10$ for NEK BHs. Contours are constant values of the logarithm of the continued fraction in the complex plane; darker shading indicates values near zero. The $+$ symbols are the ZDM predictions, a $\times$ is the lowest-overtone WKB prediction from [15], and the box is centered at the WKB prediction from Eq. (5). No branching is observed for modes with $m=8$. Note that the closed contours with light shading have large values and do not correspond to any QNM. Further discussion of the figure is in the text.