Kerr Quasinormal Modes

• Kerr quasinormal modes are defined as the damped resonances of linear perturbations in rotating black holes with discrete complex frequencies emerging from specific ingoing and outgoing boundary conditions.
• The analysis leverages the Teukolsky formalism and separability, linking mode labels (l, m, n) to underlying geometric structures and symmetry operators such as Killing tensors.
• Computational techniques like Leaver’s continued-fraction method and hyperboloidal slicing enable accurate determination of mode spectra and perturbative deformations in both near-extremal and standard regimes.

Kerr quasinormal modes are the damped resonances of linear perturbations of the Kerr black hole. In the traditional separated description, they are solutions with time dependence $e^{-i\omega t+i m\phi}$ that are purely ingoing at the future event horizon and purely outgoing at spatial infinity; the allowed complex frequencies form a discrete spectrum $\omega_{\ell m n}$ labeled by angular indices and overtone number. In rigorous formulations, the same objects arise as poles of the meromorphically continued cutoff resolvent, or as isolated eigenvalues of the infinitesimal generator of time translations on asymptotically hyperboloidal slices (Mark et al., 2014, Stucker, 2024, Gajic et al., 2024).

1. Spectral definition and boundary-value problem

In Boyer–Lindquist coordinates $(t,r,\theta,\phi)$, a spin-$s$ perturbation is separated as

$$\psi_s(t,r,\theta,\phi)=e^{-i\omega t + im\phi}\;{}_sS_{\ell m}(\theta;a\omega)\; {}_sR_{\ell m}(r;\omega,a).$$

The quasinormal-mode boundary conditions are imposed on the radial factor: near the future event horizon $r\to r_+$, the solution is purely ingoing,

$${}_sR(r)\;\propto\;(r-r_+)^{-s-i\sigma_+},\qquad \sigma_+\equiv\frac{\omega-m\Omega_H}{4\pi T_H},$$

while at spatial infinity $r\to\infty$ it is purely outgoing,

${}_sR(r)\;\propto\;r^{-1-2s}\,e^{+i\omega r}.$

These two conditions turn the spectral problem into an eigenvalue problem for discrete complex $\omega$ (Mark et al., 2014).

The rigorous resolvent definition replaces the heuristic outgoing/ingoing prescription by Fredholm and complex-scaling constructions. Writing

$\omega_{\ell m n}$0

one defines Kerr quasinormal modes as the discrete set

$\omega_{\ell m n}$1

equivalently as the values of $\omega_{\ell m n}$2 for which $\omega_{\ell m n}$3 on the scaled spaces. In this framework the definition is independent of the scaling angle $\omega_{\ell m n}$4 and the cutoff $\omega_{\ell m n}$5, the poles are discrete with finite multiplicity, there is a high-energy spectral gap, and there is no accumulation at $\omega_{\ell m n}$6 in the physical sheet (Stucker, 2024).

A complementary hyperboloidal formulation identifies quasinormal modes as isolated eigenvalues of the infinitesimal generator $\omega_{\ell m n}$7 of the time-evolution semigroup on Hilbert spaces with finite Sobolev regularity in bounded regions and Gevrey regularity at null infinity. In the open sector

$\omega_{\ell m n}$8

the spectrum is purely discrete and independent of the auxiliary parameters $\omega_{\ell m n}$9. This framework also bridges the resolvent and generator notions of Kerr quasinormal modes (Gajic et al., 2024).

2. Teukolsky separation, mode labels, and hidden structure

The separated Kerr problem is governed by the Teukolsky formalism. For spin weight $(t,r,\theta,\phi)$0, the angular equation is

$(t,r,\theta,\phi)$1

and the radial equation is

$(t,r,\theta,\phi)$2

with

$(t,r,\theta,\phi)$3

The usual labels are $(t,r,\theta,\phi)$4, where $(t,r,\theta,\phi)$5, $(t,r,\theta,\phi)$6, and $(t,r,\theta,\phi)$7 is the overtone number (Mark et al., 2014).

This separability is tied to the Petrov type D character of Kerr. Petrov type D guarantees separability of the Teukolsky equation into an ODE in $(t,r,\theta,\phi)$8 and an ODE in $(t,r,\theta,\phi)$9, and it also underlies the existence of a second-rank Killing tensor $s$0 and an infinite family of differential symmetry operators built from powers of $s$1. In the Weyl-scalar formulation, the same structure leads to commuting symmetry operators $s$2, $s$3, and $s$4, and hence to infinitely many conserved bilinear forms (Green et al., 2022).

A recurrent simplification is to identify Kerr quasinormal modes only with separated Teukolsky modes. The hyperboloidal-generator framework makes a narrower statement: regularity quasinormal modes need not lie in the separated form

$s$5

although when they do, they recover the usual $s$6 (Gajic et al., 2024).

3. Conserved bilinear forms, orthogonality, and excitation coefficients

For spin-$s$7 Weyl-scalar perturbations, a conserved bilinear structure can be built directly from the adjoint Teukolsky equation. If $s$8 and $s$9, the current

$$\psi_s(t,r,\theta,\phi)=e^{-i\omega t + im\phi}\;{}_sS_{\ell m}(\theta;a\omega)\; {}_sR_{\ell m}(r;\omega,a).$$0

is divergence-free. This gives the conserved bilinear form

$$\psi_s(t,r,\theta,\phi)=e^{-i\omega t + im\phi}\;{}_sS_{\ell m}(\theta;a\omega)\; {}_sR_{\ell m}(r;\omega,a).$$1

Using the involutive isometry $$\psi_s(t,r,\theta,\phi)=e^{-i\omega t + im\phi}\;{}_sS_{\ell m}(\theta;a\omega)\; {}_sR_{\ell m}(r;\omega,a).$$2, which swaps the two real principal null directions, one defines

$$\psi_s(t,r,\theta,\phi)=e^{-i\omega t + im\phi}\;{}_sS_{\ell m}(\theta;a\omega)\; {}_sR_{\ell m}(r;\omega,a).$$3

This bilinear form is symmetric, independent of $$\psi_s(t,r,\theta,\phi)=e^{-i\omega t + im\phi}\;{}_sS_{\ell m}(\theta;a\omega)\; {}_sR_{\ell m}(r;\omega,a).$$4, and invariant under $$\psi_s(t,r,\theta,\phi)=e^{-i\omega t + im\phi}\;{}_sS_{\ell m}(\theta;a\omega)\; {}_sR_{\ell m}(r;\omega,a).$$5- and $$\psi_s(t,r,\theta,\phi)=e^{-i\omega t + im\phi}\;{}_sS_{\ell m}(\theta;a\omega)\; {}_sR_{\ell m}(r;\omega,a).$$6-flows (Green et al., 2022).

For Kerr quasinormal modes, the naive inner product diverges because each mode grows exponentially as $$\psi_s(t,r,\theta,\phi)=e^{-i\omega t + im\phi}\;{}_sS_{\ell m}(\theta;a\omega)\; {}_sR_{\ell m}(r;\omega,a).$$7 or $$\psi_s(t,r,\theta,\phi)=e^{-i\omega t + im\phi}\;{}_sS_{\ell m}(\theta;a\omega)\; {}_sR_{\ell m}(r;\omega,a).$$8. The renormalized construction analytically continues the integrand into the complex $$\psi_s(t,r,\theta,\phi)=e^{-i\omega t + im\phi}\;{}_sS_{\ell m}(\theta;a\omega)\; {}_sR_{\ell m}(r;\omega,a).$$9-plane, or equivalently the $r\to r_+$0-plane, and deforms the real $r\to r_+$1-integration into a “snake” contour $r\to r_+$2 chosen so that

$r\to r_+$3

so that $r\to r_+$4 decays. After this renormalization, the bilinear form is finite on quasinormal modes (Green et al., 2022).

Orthogonality then follows from time-translation invariance. Since

$r\to r_+$5

one obtains

$r\to r_+$6

Thus quasinormal modes with different $r\to r_+$7 are orthogonal. In Schwarzschild, $r\to r_+$8, this further factorizes into a Kronecker $r\to r_+$9 by the usual spherical-harmonic orthogonality in $${}_sR(r)\;\propto\;(r-r_+)^{-s-i\sigma_+},\qquad \sigma_+\equiv\frac{\omega-m\Omega_H}{4\pi T_H},$$0 (Green et al., 2022).

The same bilinear form yields exact excitation coefficients. If an arbitrary perturbation $${}_sR(r)\;\propto\;(r-r_+)^{-s-i\sigma_+},\qquad \sigma_+\equiv\frac{\omega-m\Omega_H}{4\pi T_H},$$1 has quasinormal contribution

$${}_sR(r)\;\propto\;(r-r_+)^{-s-i\sigma_+},\qquad \sigma_+\equiv\frac{\omega-m\Omega_H}{4\pi T_H},$$2

then

$${}_sR(r)\;\propto\;(r-r_+)^{-s-i\sigma_+},\qquad \sigma_+\equiv\frac{\omega-m\Omega_H}{4\pi T_H},$$3

A second representation in terms of the derivative of the scaled Wronskian and the projected initial-data source is shown to coincide with this projection formula by a boundary-term identity relying on the $${}_sR(r)\;\propto\;(r-r_+)^{-s-i\sigma_+},\qquad \sigma_+\equiv\frac{\omega-m\Omega_H}{4\pi T_H},$$4-intertwining. A plausible implication is that the bilinear structure furnishes a natural starting point for nonlinear quasinormal-mode coupling in Kerr (Green et al., 2022).

4. Geometric optics, analyticity, and microlocal structure

In the eikonal regime $${}_sR(r)\;\propto\;(r-r_+)^{-s-i\sigma_+},\qquad \sigma_+\equiv\frac{\omega-m\Omega_H}{4\pi T_H},$$5, Kerr quasinormal modes are linked to unstable circular photon orbits. At leading order,

$${}_sR(r)\;\propto\;(r-r_+)^{-s-i\sigma_+},\qquad \sigma_+\equiv\frac{\omega-m\Omega_H}{4\pi T_H},$$6

where $${}_sR(r)\;\propto\;(r-r_+)^{-s-i\sigma_+},\qquad \sigma_+\equiv\frac{\omega-m\Omega_H}{4\pi T_H},$$7 is the coordinate-time orbital frequency of the relevant null orbit and $${}_sR(r)\;\propto\;(r-r_+)^{-s-i\sigma_+},\qquad \sigma_+\equiv\frac{\omega-m\Omega_H}{4\pi T_H},$$8 is its Lyapunov exponent. For equatorial $${}_sR(r)\;\propto\;(r-r_+)^{-s-i\sigma_+},\qquad \sigma_+\equiv\frac{\omega-m\Omega_H}{4\pi T_H},$$9 modes, the circular photon radii are

$r\to\infty$0

with

$r\to\infty$1

Dolan’s expansion goes beyond the leading WKB order, giving a spin-dependent $r\to\infty$2 correction for equatorial modes and a corresponding closed form for polar $r\to\infty$3 modes (Dolan, 2010).

The geometric-optics picture does not replace the full spectral problem, but it organizes a large part of the lightly damped spectrum and provides a concrete interpretation of both the oscillation frequency and the damping rate. In this regime, the “ring-down” frequencies are controlled by photon trapping, and the decay is controlled by the instability of that trapping (Dolan, 2010).

At the level of mode functions, a separate microlocal result establishes real analyticity of quasinormal modes in subextremal Kerr and Kerr–de Sitter. For a wave operator

$r\to\infty$4

the bicharacteristic flow associated to the stationary operator has a stable radial point source/sink structure with respect to a suitable horizon-generating Killing field, rather than merely a generalized normal source/sink structure. Near the event horizon one uses

$r\to\infty$5

which becomes null on $r\to\infty$6. By combining this structure with the radial-point theorem of Galkowski and Zworski, quasinormal-mode solutions are shown to extend real-analytically across the horizons (Petersen et al., 2021).

5. Nearly extremal Kerr: zero-damping and damped branches

As the spin approaches extremality, $r\to\infty$7, the Kerr spectrum develops a distinctive bifurcation. Defining $r\to\infty$8, one finds two families: zero-damping modes (ZDMs), whose decay rates vanish in the extremal limit, and damped modes (DMs), whose damping remains finite. For corotating perturbations with $r\to\infty$9, the ZDM frequencies satisfy

${}_sR(r)\;\propto\;r^{-1-2s}\,e^{+i\omega r}.$0

where

${}_sR(r)\;\propto\;r^{-1-2s}\,e^{+i\omega r}.$1

Thus ${}_sR(r)\;\propto\;r^{-1-2s}\,e^{+i\omega r}.$2 and ${}_sR(r)\;\propto\;r^{-1-2s}\,e^{+i\omega r}.$3 as ${}_sR(r)\;\propto\;r^{-1-2s}\,e^{+i\omega r}.$4 (Yang et al., 2012).

The DMs are captured by a WKB analysis in which the decay rate stays finite as ${}_sR(r)\;\propto\;r^{-1-2s}\,e^{+i\omega r}.$5. Introducing

${}_sR(r)\;\propto\;r^{-1-2s}\,e^{+i\omega r}.$6

Yang et al. derive a critical value

${}_sR(r)\;\propto\;r^{-1-2s}\,e^{+i\omega r}.$7

For ${}_sR(r)\;\propto\;r^{-1-2s}\,e^{+i\omega r}.$8, only ZDMs exist; for ${}_sR(r)\;\propto\;r^{-1-2s}\,e^{+i\omega r}.$9, ZDMs and DMs coexist (Yang et al., 2012).

The underlying reason is visible in the effective radial potential at extremality. Setting $\omega$0 and $\omega$1, the eikonal potential is

$\omega$2

If $\omega$3, the global maximum lies on the horizon, producing near-horizon ZDMs. If $\omega$4, a distinct barrier peak forms outside the horizon, producing DMs. As the spin decreases away from extremality, the two branches merge continuously into the usual single Kerr ladder (Yang et al., 2012).

A further consequence of the nearly extremal spectrum is an early-time inverse-law ringdown. Because the ZDM overtones have nearly equal real parts and nearly equally spaced imaginary parts, a coherent sum yields

$\omega$5

For $\omega$6, the early-time decay becomes

$\omega$7

Only at later times does the least-damped overtone restore purely exponential decay (Yang et al., 2013).

6. Numerical computation, full-PDE approaches, and perturbative deformations

The classical computational approach is Leaver’s continued-fraction method. One writes the radial solution as a Frobenius series satisfying the quasinormal boundary conditions, obtaining a three-term recurrence

$\omega$8

and the quasinormal-mode condition

$\omega$9

The angular equation is solved simultaneously through its own continued fraction for the separation constant. This remains the reference ODE-based method for Kerr and for many related spacetimes (Konoplya et al., 2013).

Several recent methods reformulate Kerr quasinormal modes as higher-dimensional eigenvalue problems. A hyperboloidal $\omega_{\ell m n}$00-mode approach rewrites the Teukolsky equation as a genuine $\omega_{\ell m n}$01 elliptic eigenvalue problem in $\omega_{\ell m n}$02, with the compactified horizon at $\omega_{\ell m n}$03 and future null infinity at $\omega_{\ell m n}$04. The eigenvalues are extracted directly from a block-matrix discretization,

$\omega_{\ell m n}$05

so that $\omega_{\ell m n}$06 follows without root-finding. In both radial-fixing and Cauchy-horizon-fixing gauges, the spectra show exponential convergence, and the strong gradients seen near extremality in the radial-fixing gauge are shown to be coordinate artefacts rather than physical divergences (Assaad et al., 4 Jun 2025).

A separate full-metric method works directly with the linearized Einstein equations on Kerr. After compactifying

$\omega_{\ell m n}$07

and expanding the regularized perturbation functions in Chebyshev polynomials and associated Legendre functions, the problem becomes a quadratic matrix eigenvalue system,

$\omega_{\ell m n}$08

For the $\omega_{\ell m n}$09-led and $\omega_{\ell m n}$10-led sectors, this reproduces the Kerr fundamental modes with relative error $\omega_{\ell m n}$11 for $\omega_{\ell m n}$12, remaining below $\omega_{\ell m n}$13 for $\omega_{\ell m n}$14 (Blázquez-Salcedo et al., 2023).

A more elementary discretization, formulated originally for scalar perturbations, converts the separated radial and angular equations into homogeneous matrix systems

$\omega_{\ell m n}$15

with the spectral conditions

$\omega_{\ell m n}$16

This matrix method is accurate for Kerr scalar modes and extends in a straightforward way to spin-weighted perturbations by replacing the scalar equations with the appropriate Teukolsky forms (Lin et al., 2017).

Because Kerr quasinormal eigenfunctions are not square-integrable on the real radial axis, perturbation theory for deformed backgrounds requires a nonstandard inner product. A contour prescription in the complex $\omega_{\ell m n}$17-plane yields the quantum-mechanics-style first-order shift formula

$\omega_{\ell m n}$18

which has been used for small deformations of Kerr and, in particular, for weakly charged Kerr–Newman spacetimes (Zimmerman et al., 2014). In the weak-charge expansion $\omega_{\ell m n}$19, the corresponding first-order correction is

$\omega_{\ell m n}$20

and the calculated corrections do not reveal unstable modes in the weakly charged regime (Mark et al., 2014).

Kerr quasinormal modes thus occupy a dual role. They are, on the one hand, a rigorously defined resonance spectrum of a subextremal rotating black hole. On the other hand, they are a computationally accessible and geometrically structured family of modes whose separability, orthogonality, eikonal interpretation, near-extremal bifurcation, and perturbative deformations continue to organize both analytic and numerical work on rotating black-hole perturbations.