Zero-Damped and Damped Mode Families

The Kerr quasinormal spectrum becomes singular near extremality. One family localizes toward the throat and has a damping rate that vanishes with surface gravity; another remains associated with an exterior potential barrier and retains finite damping. These zero-damped and damped modes can coexist, but their existence domains and limiting wavefunctions differ. Neither family is an exponentially growing instability: with the convention $e^{-i\omega t+im\phi}$, every finite-subextremal quasinormal frequency discussed here remains in the lower half-plane.

Helpful background. Black-Hole Geometry and Mechanics supplies Kerr horizons and photon regions; PDEs and Green Functions for Gravitation supplies resonances and singular spectral limits; Formulations of General Relativity supplies the Teukolsky equation; Events, Intervals, and Minkowski Geometry fixes decay signs; Black-Hole Perturbation Theory supplies quasinormal boundary conditions.

Two resonance families

Write $\mu=GM$ and define

$$\sigma=\frac{r_+-r_-}{r_+}=4\mu\kappa.$$

A Kerr quasinormal mode is ingoing at the future event horizon and outgoing at future null infinity. Its frequency has $\operatorname{Im}\omega<0$ when it decays under $e^{-i\omega t}$. Near extremality the spectrum separates as follows.

Family	Frequency as $\kappa\to0$	Radial support	Existence pattern
Zero-damped modes (ZDMs)	$\operatorname{Re}\omega\to m/(2\mu)$ and $-\operatorname{Im}\omega=O(\kappa)$	Increasingly concentrated in the near-horizon region	Present for the corotating indices covered by the matched analysis; in the Yang et al. convention, all $\ell$ and $m\geq0$
Damped modes (DMs)	$-\operatorname{Im}\omega\to\gamma_{\ell mn}>0$	Exterior potential peak remains separated from the horizon	All counterrotating sectors and only a subset of corotating sectors

The classification is about a limiting family, not just one small imaginary part. A moderately spinning Kerr mode can connect continuously to either branch as parameters vary. When both branches exist, they merge into the ordinary single spectrum away from the near-extremal regime. The bifurcation and its effective-potential interpretation are established in Yang et al. 2013, “Branching,” §§ II–IV, pp. 2–4.

Zero-damped scaling

The ZDM calculation uses three correlated assumptions: $\sigma\ll1$, $r-r_+=O(\sigma r_+)$ in the near region, and $\omega-m\Omega_H=O(\kappa)$. Introduce the finite detuning

$$\varpi=\frac{\omega-m\Omega_H}{\kappa}.$$

Matching the near hypergeometric solution to a far confluent-hypergeometric solution produces a gamma-function resonance condition. In the site’s angular convention, define

$$\delta^2=\frac{7m^2}{4}-\left(s+\frac12\right)^2-A_{s\ell m}(m/2), \qquad \epsilon=1-\frac{a}{\mu}.$$

Here $A_{s\ell m}(c)$ is the angular Teukolsky eigenvalue, with spherical limit $\ell(\ell+1)-s(s+1)$. For the following generic pole approximation take $m\gt0$, real $\delta\gt0$, fixed overtone $n$, and a small pole correction $\eta$. The frequency predicted by the first Gamma-pole expansion is

$$\mu(\omega-m\Omega_H)\simeq\sqrt\epsilon \left[\frac{m-\delta-i(n+1/2)}{\sqrt2}+\eta(\epsilon)\right].$$

The exact surface gravity is

$$\mu\kappa=\frac{\sqrt{2\epsilon-\epsilon^2}} {2(1+\sqrt{2\epsilon-\epsilon^2})}, \qquad \frac{\sqrt\epsilon}{\mu\kappa}\longrightarrow\sqrt2.$$

Consequently, at this retained order,

$$\begin{aligned} \varpi&\simeq m-\delta-i(n+1/2)+\sqrt2\,\eta(\epsilon),\\ \operatorname{Im}\varpi&\simeq-(n+1/2)+\sqrt2\,\operatorname{Im}\eta. \end{aligned}$$

The approximate equality denotes the matched and small-$\eta$ approximations; it is not a uniform error estimate. In particular, discarding a small $\eta$ is not generally a little-$o(1)$ statement as $\epsilon\to0$. At fixed limiting angular parameters the leading correction has the form $\eta(\epsilon)=C_{\ell mn}\exp[-i\delta\log\epsilon]$, where the constant $C_{\ell mn}$ is a product of Gamma functions. Its modulus is constant for real $\delta$. For nonzero $C_{\ell mn}$, the two sequences

$$\epsilon_j=e^{-2\pi j/\delta},\qquad \epsilon'_j=e^{-(2\pi j+\pi)/\delta}$$

approach zero but give opposite correction phases. Thus the scaled frequency can remain bounded without approaching a single coefficient. The Gamma-pole calculation and its oscillatory correction are in Yang et al. 2013, “Spectrum bifurcation,” Eqs. (3.17)–(3.20), pp. 8–9 (PDF).

When the small correction does not cancel the positive damping coefficient, the damping time grows as $\kappa^{-1}$ at fixed $n$. If $\delta^2\lt0$, then choose $\delta=i\rho$ with $\rho\gt0$, and the leading term proportional to $-\delta$ also contributes to $\operatorname{Im}\varpi$; the coefficient $n+1/2$ is not universal across that regime even though the damping remains $O(\kappa)$ in the stated near-extremal branch. The coalescent $\delta=0$ case and the $m=0$ case require separate limiting bases; they are not obtained by substituting into a formula containing $\log m$ or separately divergent Gamma coefficients. See Yang et al. 2013, “Spectrum bifurcation,” § III.C–D, pp. 9–10 (PDF).

The radial localization supplies a geometric check. In the eikonal picture, the relevant potential peak approaches the horizon, its tortoise-coordinate curvature tends to zero, and the associated decay rate vanishes. In throat coordinates, the same modes retain a finite frequency and profile. At fixed Boyer–Lindquist radius outside the throat, their normalized wavefunction can instead become small or singular depending on which amplitude is held fixed. Mode normalization and observation region must accompany any statement about “large” ZDM excitation.

Damped modes and the branching boundary

Damped modes are controlled by a potential peak that remains outside the horizon. Their imaginary parts approach nonzero negative values as $\kappa\to0$, so their lifetimes do not inherit the throat scale. In the eikonal limit, let

$$\chi_m=\frac{m}{\ell+1/2}.$$

The Yang et al. analysis finds a critical value near $0.74$: corotating sectors with smaller $\chi_m$ can support an exterior peak and hence a damped branch, while sufficiently large $\chi_m$ has only the zero-damped branch in that limit. Finite-$\ell$ boundaries are determined from the angular eigenvalue and numerical spectrum, not by treating $0.74$ as an exact universal constant. Counterrotating modes belong to the damped family under the positive-real-frequency convention.

This distinction is independently testable. For a sequence of spins, compute the frequencies with a converged continued fraction or spectral boundary-value method. A ZDM gives a bounded scaled sequence $(\omega-m\Omega_H)/\kappa$ and moves radially into $x=(r-r_+)/r_+=O(\sigma)$. A DM gives a finite nonzero limit of $-\operatorname{Im}\omega$ and an exterior peak. Tracking only the imaginary part at one spin can misclassify a long-lived ordinary resonance as a ZDM.

No new continued-fraction or spectral sequence is computed on this page. The branch classification and scaling statements above are source-backed analytic results, while the preceding paragraph states the numerical test that a fresh quantitative table would still have to pass. In particular, the page does not claim an independently rerun error bound for the matched formula.

Pole accumulation and time response

As $\kappa\to0$, the ZDM poles crowd near the synchronous real frequency. Their spacing and damping are both $O(\kappa)$. At times with $1\ll t/\mu\ll1/(\mu\kappa)$, many overtones can contribute coherently and produce an apparent power-law ringdown. Within the QNM contribution, the least-damped exponential subsequently takes over at fixed $\kappa\gt0$. The explicit scalar calculation assumes source and observer at large radius, keeps the leading near-extremal and small-pole-shift terms, and uses time measured after propagation from the source; see Yang et al. 2013, “Spectrum bifurcation,” § V.A, Eqs. (5.1)–(5.12), pp. 16–17 (PDF). This is not the ultimate asymptotic of the complete asymptotically flat signal. A nonzero branch tail eventually dominates exponentially damped poles, as the source also notes on Yang et al. 2013, p. 2 (PDF).

A simple model makes the distinction explicit without claiming Kerr residues. For $t\gt0$ and $\kappa\gt0$, sum equally weighted poles:

$$g_\kappa(t)=\kappa\sum_{n=0}^{\infty}e^{-\kappa(n+1/2)t} =\frac{\kappa}{2\sinh(\kappa t/2)}.$$

It behaves as $1/t$ when $\kappa t\ll1$, and as $\kappa e^{-\kappa t/2}$ when $\kappa t\gg1$. But adding a nonzero $Ct^{-3}$ term changes the ultimate behavior, since $t^3g_\kappa(t)\to0$ at fixed $\kappa$. The exponent three is only a toy choice; a physical tail requires its field, data and observation region.

The limiting pole accumulation may become a branch singularity in an extremal frequency-domain Green function. It does not follow that the poles form a complete basis, nor that their sum alone supplies prompt propagation or the late-time asymptotically flat tail. The limits $\kappa\to0$, $t\to\infty$, and $\omega\to m\Omega_H$ must be specified before replacing a pole sequence by a continuum expression.

The NHEK weights underlying the ZDM branch are derived in Field Equations in Near-Horizon Extreme Kerr. Their connection to the exterior is constructed in Matched Asymptotic Expansions Across the Throat.

Exercises

Suppose a ZDM and a DM have

$$\operatorname{Im}\omega_{\mathrm Z} =-\kappa\left(n+\frac12\right), \qquad \operatorname{Im}\omega_{\mathrm D}=-\gamma,$$

with $\gamma>0$ fixed and $\operatorname{Re}\omega_{\mathrm Z}\to \omega_*\ne0$ (for example $m\gt0$ at fixed $\mu$). Find their amplitude decay times and explain why a diverging ZDM quality factor is not an instability.

Solution

Because $e^{-i\omega t}$ has amplitude $e^{\operatorname{Im}\omega\,t}$,

$$\tau_{\mathrm Z} =\frac{1}{\kappa(n+1/2)}, \qquad \tau_{\mathrm D}=\frac{1}{\gamma}.$$

At fixed overtone, $\tau_{\mathrm Z}$ diverges as $\kappa^{-1}$, whereas the DM lifetime stays finite. The ZDM quality factor $Q\sim\operatorname{Re}\omega/(2\lvert\operatorname{Im}\omega\rvert)$ therefore diverges. Nevertheless, $\operatorname{Im}\omega_{\mathrm Z}<0$ for every $\kappa>0$, so the amplitude decays rather than grows. The divergent lifetime signals a nonuniform extremal limit and coherent long-lived response, not a pole in the upper half-plane.

References

• Yang, H., Zhang, F., Zimmerman, A., Nichols, D. A., Berti, E., and Chen, Y. (2013). “Branching of quasinormal modes for nearly extremal Kerr black holes.” Physical Review D 87, 041502(R). DOI.
• Yang, H., Zimmerman, A., Zenginoğlu, A., Zhang, F., Berti, E., and Chen, Y. (2013). “Quasinormal modes of nearly extremal Kerr spacetimes: Spectrum bifurcation and power-law ringdown.” Physical Review D 88, 044047. DOI.