Ringdown Quasinormal Mode

What "a black hole rings" actually means

A black hole has no surface. There is no skin to vibrate, no air to resonate, no medium at all — only curved vacuum spacetime around a one-way membrane. So how can a black hole ring?

The answer is that the geometry itself oscillates. After any disturbance — a merger, an in-falling test mass, a quantum kick — the metric around a black hole deviates from its stationary Kerr form. General relativity is a hyperbolic theory; perturbations propagate as gravitational waves and radiate to infinity. But not every perturbation radiates at every frequency. The vacuum Kerr spacetime has a discrete spectrum of resonant modes — characteristic complex frequencies at which it prefers to ring. Each is a damped sinusoid of strain at large distance:

h(t) = h₀ e^(-t/τ) cos(2π f t + φ)

The real part of the frequency, f, is the pitch. The imaginary part, packaged as a damping time τ = 1/(2π Im ω), is the rate at which the ringing decays — entirely because energy is leaking out as gravitational waves. There is no friction, no dissipation, no medium; the damping is pure gravitational radiation reaction.

This is why they are "quasi-normal" rather than normal. A normal mode of an ordinary system — a guitar string, a wineglass — oscillates forever at a real frequency, because the system is closed. A black hole's perturbations have a one-way boundary condition: outgoing at infinity, ingoing at the horizon, no reflection. Energy is constantly drained from the oscillation, so the eigenvalues are intrinsically complex. The system rings, but it cannot ring forever.

Why a fresh merger remnant must ring

Take two black holes orbiting in a binary. As they spiral together they emit gravitational waves, lose orbital energy, and finally plunge through their mutual horizons in a fraction of a millisecond. What is left, in the immediate aftermath, is a single compact object — but it is not yet a Kerr black hole.

By the no-hair theorem, the only stationary, axially symmetric, asymptotically flat, vacuum solution of general relativity with a regular horizon is the Kerr metric, parameterised by exactly two numbers: mass M and dimensionless spin a = Jc/(GM²) ∈ [0, 1]. Everything else — quadrupole, octupole, hexadecapole moments, lopsidedness, "tidal bulges" inherited from the merger — is "hair," and any hair must be radiated away.

The remnant has hair. Its multipole moments do not match Kerr's. Its surface (well, its horizon) is wrinkled. Mathematically, the perturbation away from Kerr can be expanded in the natural basis of QNMs, and the remnant rings down by radiating each excited mode's worth of hair as gravitational waves. Within a few damping times — typically ~10 ms for a 60 M☉ merger — the remnant has shed essentially all the hair it inherited from the merger and has become a clean Kerr black hole, completely characterised by M and a.

The (l, m, n) labelling and why (2, 2, 0) wins

Quasinormal modes of a Kerr black hole are labelled by three integers:

• l ≥ 2 — the angular multipole index. For gravitational radiation, dipole emission is forbidden by conservation of momentum and quadrupole emission (l = 2) is the lowest allowed. Higher l means more nodes on the sky.
• −l ≤ m ≤ l — the azimuthal index, governing rotation around the spin axis.
• n ≥ 0 — the overtone index. n = 0 is the longest-lived (smallest damping rate); higher overtones decay faster.

Among all of these, the (l, m, n) = (2, 2, 0) mode dominates the post-merger waveform of nearly every observed compact binary, for one simple reason: the inspiral and merger themselves are overwhelmingly quadrupolar. The orbital motion is a quadrupolar mass distribution rotating at orbital frequency, so the strongest excitation of the remnant is the matching quadrupolar shape, which couples most strongly to (2, 2, 0).

Higher modes are present, just exponentially fainter. The (2, 2, 1) overtone, the first overtone of the dominant mode, decays roughly three times faster than (2, 2, 0) and so is short-lived. The (3, 3, 0) mode is suppressed by mass-ratio (it vanishes for equal-mass binaries by symmetry, so non-equal-mass mergers like GW190521 are the best laboratories for it). The (4, 4, 0), (2, 1, 0), and other modes contribute at the few-percent level. Black hole spectroscopy aims to dig these subdominant modes out of the data.

The mass-only scaling for Schwarzschild

For a non-spinning (Schwarzschild) black hole of mass M, every QNM frequency and damping time depends on M alone. The dominant mode numerical values, expressed in convenient units, are

f_(2,2,0)   ≈ 12 kHz × (M☉ / M)
τ_(2,2,0)   ≈ 0.55 ms × (M / M☉)

This is dimensional. The only length scale in vacuum general relativity is r_g = GM/c²; the only timescale is r_g/c = GM/c³. A black hole "rings" on a timescale of light-crossing the gravitational radius — so f ~ c³/(GM). Plug in c = 3 × 10⁸ m/s, G = 6.674 × 10⁻¹¹ m³ kg⁻¹ s⁻², and M = 1 M☉ = 1.989 × 10³⁰ kg, and you get f ~ 32 kHz; the numerical coefficient brings it down to roughly 12 kHz for the (2, 2, 0) mode.

Concrete examples follow directly:

Remnant	Mass	f_(2,2,0)	τ_(2,2,0)	Detector band
Hypothetical primordial BH	10⁻¹ M☉	~120 kHz	0.055 ms	None (above all current detectors)
Stellar-mass remnant	10 M☉	~1.2 kHz	5.5 ms	LIGO/Virgo/KAGRA
GW150914-class remnant	60 M☉	~200 Hz	33 ms	LIGO/Virgo
GW190521-class remnant	150 M☉	~80 Hz	83 ms	LIGO/Virgo, low-frequency edge
Intermediate-mass BH	10⁴ M☉	~1.2 Hz	5.5 s	Gap band (CE/ET reach)
SMBH binary remnant	10⁶ M☉	~12 mHz	9 minutes	LISA
Most-massive AGN merger	10⁹ M☉	~12 μHz	~6 days	Pulsar timing arrays (in principle)

The takeaway is that QNM frequencies span twelve decades — from microhertz (SMBH mergers) to kilohertz (stellar-mass mergers) to (extrapolated) hundreds of kilohertz for primordial black holes. Each band has its own detector technology, and each detector technology probes a different mass regime.

Adding spin: the full f(M, a) and τ(M, a)

For a rotating Kerr black hole, both f and τ pick up spin corrections. The dimensionless frequency Mω = M(2πf − i/τ) for the (2, 2, 0) mode rises smoothly with prograde spin a and falls (and damps more strongly) for retrograde spin. Useful fitting formulae (Berti, Cardoso, Will 2006) give Mω₂₂₀(a) and the quality factor Q₂₂₀(a) = πfτ at percent-level accuracy:

M ω_(2,2,0) (a) ≈ 1.5251 - 1.1568 (1-a)^(0.1292)
Q_(2,2,0)   (a) ≈ 0.7000 + 1.4187 (1-a)^(-0.4990)

For a = 0 (Schwarzschild): Mω ≈ 0.374, Q ≈ 2.12 — corresponding to f × M ≈ 5.95 × 10⁻⁶ s × (c³/G), and so f ≈ 12 kHz/(M/M☉). For a = 0.9: Mω ≈ 0.672, the frequency is roughly twice as high at the same mass, and Q ≈ 4.5, so the mode rings for more than twice as many cycles before decaying. For near-extremal Kerr (a → 1), Q diverges logarithmically: the (2, 2, 0) mode becomes very long-lived, which is one route to extracting tight constraints on a from a long-duration ringdown.

Crucially, both f and τ depend on the same two parameters (M, a). So if you measure both f and τ from a single ringdown, you over-determine (M, a) — you have one redundant number, which is essentially a consistency check. To test no-hair you want a second mode: that gives you two more numbers (f₂, τ₂) for the same (M, a), so two redundant constraints. Three modes gives four. The more modes, the stronger the test.

How to compute the spectrum: the Leaver method

Linear perturbations of Kerr separate (Teukolsky 1972) into an angular and a radial Heun-type ODE. The radial equation has regular singular points at the inner and outer horizons and an irregular singular point at infinity; QNMs satisfy purely outgoing boundary conditions at infinity and ingoing at the horizon. There is no closed-form solution.

In 1985 Edward Leaver published an extraordinarily compact numerical method based on a three-term recurrence and continued fractions. Expand the radial Teukolsky wavefunction R(r) as a power series whose coefficients aₙ satisfy

α_n a_{n+1} + β_n a_n + γ_n a_{n-1} = 0

where αₙ, βₙ, γₙ are explicit functions of (l, m, ω, a). The series converges if and only if the eigenfrequency ω lies on a specific branch of a continued-fraction equation:

β_0 - (α_0 γ_1) / (β_1 - (α_1 γ_2) / (β_2 - ... )) = 0

This is the Leaver equation. Numerically, you fix (l, m, a) and search the complex ω-plane for roots of the continued fraction; the root of largest imaginary part is the fundamental mode, the next is the first overtone, and so on. Modern implementations (Berti's catalogue, Leo Stein's qnm Python package, the BHPToolkit) tabulate hundreds of modes to many decimal places.

Alternative methods include WKB approximation (Schutz-Will 1985, accurate to a few percent for low overtones; Konoplya 2003 extends to high orders), direct time-domain evolution of the Teukolsky equation, and Pöschl-Teller approximation (treats the potential as a soliton and yields analytic expressions in special limits). All agree where they overlap; Leaver remains the workhorse.

Black hole spectroscopy — turning ringdowns into GR tests

Detecting one QNM from a ringdown lets you read off (M, a) of the remnant. Detecting two or more lets you check the no-hair theorem: do all modes correspond to the same (M, a)?

Formally, model the post-merger waveform as a sum of damped sinusoids,

h(t) = Σ_{l,m,n}  A_{lmn} e^(-t/τ_{lmn}) cos(2π f_{lmn} t + φ_{lmn})

and ask: are the frequencies and damping times consistent with a single (M, a) through f_{lmn}(M, a) and τ_{lmn}(M, a) from the Leaver computation? If they are, GR is fine. If they are not, either (a) the remnant is not a Kerr black hole — perhaps it has additional structure (exotic compact object, "echoes"), or (b) general relativity itself fails in the strong field — Brans-Dicke, dynamical Chern-Simons, scalar-tensor, or other modifications would give a shifted QNM spectrum.

The practical challenge is signal-to-noise. The (2, 2, 0) mode is loud; subdominant modes are 10–100 times fainter. You need a ringdown with overall SNR ≳ 100 to extract a second mode at meaningful confidence — and most current LIGO ringdowns have SNR 5–20 in the post-merger window. GW150914 had a ringdown SNR of about 7; GW190521 around 14. The first robust two-mode detection will likely require either an unusually loud event (a nearby IMBH merger) or third-generation detectors.

GW150914 — the first ringdown ever heard

On 14 September 2015, LIGO Hanford and Livingston recorded the first direct detection of gravitational waves: GW150914, the merger of two stellar-mass black holes of (35.6, 30.6) M☉ at a luminosity distance of 410 Mpc. The signal lasted about 0.2 s in the LIGO band, ramping from 35 Hz through several hundred hertz and ending in a brief ringdown at about 250 Hz.

That ringdown was the first direct observation of black hole quasinormal mode emission. The signal was well-modelled as a single damped sinusoid corresponding to the (2, 2, 0) mode of a Kerr remnant of M ≈ 62 M☉ and a ≈ 0.67. The inferred (f, τ) matched the (M, a) implied by the inspiral parameters via the Leaver computation — the first observational confirmation that the merger product was a Kerr black hole, not some exotic alternative.

Attempts to extract a second mode from GW150914 have not yielded a confident detection. The ringdown SNR was simply too low. But the consistency of (f_(2,2,0), τ_(2,2,0)) with (M, a) constituted a clean, if minimal, test of the no-hair theorem at the level of one mode.

GW190521 — a hint of two modes?

GW190521 (May 2019) was a much more massive event: two ~85 M☉ black holes merging to form a ~150 M☉ remnant. Because frequencies scale as 1/M, the ringdown sat at ~80 Hz — lower in the detector band, longer in duration, with damping time τ ≈ 70 ms. The longer ringdown made it the most promising candidate yet for two-mode detection.

Capano et al. (2021) reanalysed GW190521 and reported tentative evidence (~2σ) for a (3, 3, 0) mode in addition to (2, 2, 0). The asymmetric mass ratio (~1:1.3) of the parent binary, if real, would suppress (2, 2, 0) only modestly while exciting (3, 3, 0) significantly. Subsequent analyses with different priors and waveform models reached similar marginal conclusions: not a confident detection, but not a clear non-detection either. The community consensus is that no robust two-mode detection has yet been claimed at high confidence.

LISA, CE, ET — the spectroscopy era

Three next-generation detectors will, in the late 2030s, change the picture entirely.

• LISA (ESA + NASA, launching ~2035): three spacecraft in a 2.5-million-km triangular constellation, sensitive to 0.1 mHz – 0.1 Hz. Its targets are supermassive black hole mergers (10⁵–10⁷ M☉) detectable across all of cosmic history with ringdown SNRs of 10² to 10⁴. Even a single such event will yield five or more cleanly resolved modes; the population over its 4-year mission could deliver hundreds.
• Cosmic Explorer (US, ~2035): a 40-km L-shaped interferometer, an order of magnitude more sensitive than Advanced LIGO across the 5 Hz – 5 kHz band. Stellar-mass merger ringdowns will routinely have SNR > 100. Two-mode detections will become standard.
• Einstein Telescope (Europe, ~2035): a 10-km triangular underground detector, similar reach to CE but with extended low-frequency band down to ~1 Hz. Together with CE, it will resolve ringdowns of every stellar-mass merger to redshift z > 1.

Together these detectors will perform black hole spectroscopy on hundreds of events. By comparing measured QNM spectra across mass and spin to the predictions of the Kerr metric, they will probe the no-hair theorem at strengths well beyond Solar System tests. Any systematic deviation — a single mode that disagrees by > 1% with the Kerr prediction at the same (M, a) — would be evidence for new physics: an exotic compact object instead of a true black hole, a modified gravity theory at the strongest fields, or both.

Worked example: predicting GW150914's ringdown frequency

Inspiral analysis of GW150914 gave a final Kerr remnant of M_f ≈ 62 M☉ and a_f ≈ 0.67. What ringdown frequency and damping time does Leaver-method theory predict for the dominant (2, 2, 0) mode?

Step 1: dimensionless frequency from the fit. Using Mω₂₂₀(a) ≈ 1.5251 − 1.1568 (1−a)^0.1292 with a = 0.67,

M ω = 1.5251 - 1.1568 × (1 - 0.67)^0.1292
   = 1.5251 - 1.1568 × 0.33^0.1292
   = 1.5251 - 1.1568 × 0.866
   = 1.5251 - 1.0019
   = 0.5232

Step 2: convert to physical frequency. ω = 2πf; M in geometric units is M (G/c³). For M = 62 M☉,

G M / c³ = 6.674e-11 × 62 × 1.989e30 / (3e8)³
        = 8.23e21 / 2.7e25
        = 3.05e-4 s
M ω = 2π f × 3.05e-4 = 0.5232
   →  f = 0.5232 / (2π × 3.05e-4)
       = 273 Hz

Step 3: damping time. Q ≈ 0.7 + 1.4187 (1−a)^(−0.499) = 0.7 + 1.4187 × (0.33)^(−0.499) ≈ 0.7 + 2.47 = 3.17. Since Q = πfτ, τ = Q/(πf) = 3.17 / (π × 273) = 3.7 ms.

So Leaver-method theory predicts f ≈ 273 Hz and τ ≈ 3.7 ms for GW150914's dominant ringdown mode. LIGO's measured values are roughly 250 Hz and 4 ms — agreement to within a few percent, set by the precision of the (M, a) inspiral inferences rather than by QNM theory itself. This is the first observational confirmation of the no-hair-theorem-consistent QNM prediction.

Beyond Kerr: scalar, vector and tensor modes

The full QNM spectrum of Kerr includes more than just gravitational (tensor) modes. Test scalar fields propagating on Kerr have their own QNMs; electromagnetic perturbations have their own. The Teukolsky equation actually handles spin-s perturbations with s = 0 (scalar), s = ±1 (electromagnetic) and s = ±2 (gravitational) by the same formalism. In modified gravity theories (e.g. Brans-Dicke, dynamical Chern-Simons), additional polarisations and extra dynamical degrees of freedom imply extra QNMs beyond the GR set. Detecting an "anomalous" mode that does not fit any (l, m, n, s = 2) GR pattern would be smoking-gun evidence for new physics.

The flip side: exotic compact objects (boson stars, gravastars, fuzzballs, firewalls) that mimic the inspiral and merger waveform of black holes have different QNM spectra in their ringdown, often with "echoes" — late-time reflections off the would-be surface. Searches for echoes in LIGO ringdowns have so far yielded no confirmed detections, with upper limits already significant constraints on certain ECO models.

Common pitfalls and subtleties

• QNMs are not normal modes. The eigenfunctions of a black hole's perturbation operator are not square-integrable on a Cauchy slice; they do not form an orthonormal basis. Decomposing an arbitrary perturbation in QNMs is well-defined only late-time, after a power-law tail (Price tail) sets in.
• The "start time" of ringdown is a choice, not a fact. Real merger waveforms transition smoothly from merger to ringdown over a few light-crossing times. Different analyses use different start times; the inferred mode amplitudes depend sensitively on this choice. Modern analyses (Isi et al. 2019, Cabero et al. 2020) fit over a range of start times to constrain this systematic.
• Overtones can mimic higher multipoles. The (2, 2, 1) overtone has nearly twice the damping rate of (2, 2, 0) but the same frequency. In low-SNR data, an overtone signal can look like a beat with another (l, m) mode. Spectroscopic claims require multi-component fits and careful systematics.
• Q diverges near extremal Kerr. As a → 1, the damping rate of (2, 2, 0) goes to zero logarithmically. The Q factor becomes very large; ringdowns of near-extremal black holes are long-lived. This is observationally favourable but theoretically delicate — the perturbation theory becomes singular at exactly a = 1.
• Confusion with the inspiral chirp. Some popular treatments label "the chirp" (the rising-frequency pre-merger phase) as "the ringdown." It is not. The chirp is inspiral; ringdown is the post-merger damped oscillation. The two have distinct frequency and time-amplitude behaviours.