Quasinormal Modes

A bell that cannot hold its note

Strike a wine glass and it sings a clean, sustained tone; the rim oscillates at a definite frequency and the note decays only slowly as friction and sound radiation bleed away the energy. A black hole, perturbed by infalling matter or by the violent collision of two other black holes, does something deeply similar — it rings. But it is a terrible bell. Its tones die within a handful of oscillations because a black hole has no rigid material to store vibrational energy; instead, every wiggle of the curved spacetime around it radiates gravitational waves to infinity and pours energy down across the event horizon, which behaves like a perfect one-way drain.

The characteristic tones it emits while settling down are the quasinormal modes. The qualifier "quasi" is the whole story: an ordinary bell has true normal modes — real, undamped eigenfrequencies of a closed, energy-conserving system. A black hole is an open system, leaking on both ends, so its modes are damped. Mathematically this turns into a single, remarkable statement: the eigenfrequencies are complex numbers, where the real part is the pitch you would hear and the imaginary part is how fast the tone fades. And here is the punchline that makes quasinormal modes one of the cleanest tests in physics: the entire set of tones is fixed by just two numbers — the black hole's mass and its spin.

The complex frequency, and what each part means

A quasinormal oscillation seen by a distant detector looks like a single decaying sinusoid:

h(t) = A e^(−ω_I t) cos(ω_R t + φ)
     = A Re[ e^(−i ω t) ],   with   ω = ω_R − i ω_I

The real part ω_R is the angular oscillation frequency — divide by 2π to get the pitch in hertz. The imaginary part ω_I sets the exponential damping; its reciprocal is the damping time, τ = 1/ω_I, after which the amplitude falls by a factor of e. Because both are nonzero, no single quasinormal mode is a true eigenstate of a Hermitian operator — the boundary conditions (purely ingoing at the horizon, purely outgoing at infinity) make the problem non-Hermitian, which is exactly why the spectrum is complex and discrete.

A useful single number is the quality factor, Q = ω_R / (2 ω_I), the number of radians of oscillation per e-folding of decay (roughly, the number of audible cycles). A church bell has Q in the thousands. A black hole's fundamental mode has Q of only about 2–4 — it manages just a few rings before going silent. That low Q is precisely why detecting a ringdown cleanly is hard: there is barely any signal to fit.

The governing equations: Regge-Wheeler, Zerilli, Teukolsky

For a non-rotating Schwarzschild black hole, a small perturbation of the metric splits by parity into an odd (axial) sector and an even (polar) sector. Tullio Regge and John Wheeler showed in 1957 that the odd sector obeys a wave equation with a fixed effective potential; Frank Zerilli derived the even-sector counterpart in 1970. Written in the "tortoise" coordinate r* (which stretches the region near the horizon out to −∞), both reduce to a Schrödinger-like form:

d²ψ/dr*²  +  [ ω² − V_ℓ(r) ] ψ  =  0

Regge-Wheeler potential (spin-2, odd parity):
V_ℓ(r) = (1 − 2GM/rc²) [ ℓ(ℓ+1)/r² − 6GM/(r³c²) ]

The potential V_ℓ(r) is a single hump — a barrier — peaked just outside the photon sphere at r = 3GM/c². A quasinormal mode is a solution that is purely ingoing at the horizon (r* → −∞) and purely outgoing at infinity (r* → +∞) at the same time. Those two outgoing/ingoing conditions can only be met for a discrete, complex set of ω — exactly the resonances of waves leaking off the potential barrier.

For a rotating Kerr black hole the metric is no longer spherically symmetric, so the simple split fails. The breakthrough was Saul Teukolsky's 1972 master equation, which miraculously separates the curvature perturbations into radial and angular factors using spin-weighted spheroidal harmonics ₋₂S_ℓm(aω, θ). The quasinormal frequencies are again the eigenvalues for which the radial wave is ingoing at the horizon and outgoing at infinity. In practice they are computed by Leaver's continued-fraction method (Edward Leaver, 1985), which remains the workhorse algorithm and is tabulated in widely used software such as the qnm Python package and the LIGO–Virgo ringdown pipeline pyRing.

The actual numbers

It is conventional to quote the dimensionless combinations Mω in geometric units (G = c = 1), where multiplying a black hole's mass turns them into real frequencies. For the Schwarzschild fundamental gravitational mode (spin-weight −2, ℓ = 2, overtone n = 0):

M ω_R ≈ 0.37367   (real part — the pitch)
M ω_I ≈ 0.08896   (imaginary part — the damping)
Q = ω_R / (2 ω_I) ≈ 2.10

To convert to SI for a black hole of mass M = m × M☉, use the characteristic frequency unit c³/(GM) and the fact that GM☉/c³ ≈ 4.925 microseconds:

f = ω_R / (2π) ≈ 0.3737 / (2π) × c³/(GM)
  ≈ 12.07 kHz × (M☉ / M)        for ℓ=2, n=0 Schwarzschild
τ = 1/ω_I ≈ 11.24 × GM/c³
  ≈ 0.0554 ms × (M / M☉)

Spin shifts these. As the dimensionless spin a* = cJ/(GM²) rises from 0 toward its extremal value of 1, the dominant prograde ℓ=2 frequency increases (the bell rings higher) and the damping slows (Q rises sharply, exceeding ~5 near a* ≈ 0.98). This spin dependence is exactly what makes the ringdown a spin-meter.

Discovery and the people who heard it first

The field has a clean lineage. Tullio Regge and John Wheeler (1957) wrote down the stability equation for Schwarzschild perturbations. C. V. Vishveshwara (1970) did the decisive numerical experiment: he scattered a Gaussian wave packet off a Schwarzschild potential and watched the response settle into a damped sinusoid — the first sighting of a quasinormal ringing, and the demonstration that black holes are stable. William Press (1971) coined the phrase "quasi-normal" and tied the ringing to the photon sphere. Frank Zerilli (1970) completed the even-parity case, and Saul Teukolsky (1972) generalised everything to Kerr. Subrahmanyan Chandrasekhar and Steven Detweiler (1975) produced the first accurate frequency tables, and Edward Leaver (1985) gave the continued-fraction method still used today.

The phenomenon stayed theoretical for 45 years. Then on 14 September 2015, LIGO recorded GW150914, the first direct detection of gravitational waves: two black holes of about 36 and 29 solar masses merged into a single ~62 M☉, a* ≈ 0.67 black hole. The final ~5 milliseconds of that signal — the part after the two holes had become one — was the long-predicted ringdown, the first time humanity heard a black hole ring. The LIGO–Virgo–KAGRA collaboration's later analyses of louder events (notably the high-mass merger GW190521) have pushed toward resolving a second mode, the goal of true black-hole spectroscopy.

Worked example: ringdown of GW150914

Take the GW150914 remnant: mass M ≈ 62 M☉, spin a* ≈ 0.67. We want the frequency and damping time of the dominant ℓ=2, m=2, n=0 mode and decide whether it lands in LIGO's band.

For a Schwarzschild hole the scaling above gives f ≈ 12.07 kHz × (M☉/M). For a* ≈ 0.67 the dimensionless real frequency rises from 0.3737 to about Mω_R ≈ 0.51, so multiply by 0.51/0.3737 ≈ 1.36:

f ≈ 12.07 kHz × (1/62) × 1.36
  ≈ 265 Hz        (LIGO is most sensitive 30–500 Hz ✓)

For the damping, Mω_I ≈ 0.083 at a* = 0.67:
τ = 1/ω_I = (GM/c³) / (M ω_I)
  = 4.925 µs × 62 / 0.083
  ≈ 3.7 ms

So the predicted ringdown sits near 250–270 Hz with a damping time of roughly 4 ms — exactly where the published GW150914 ringdown was measured. In ~4 ms at 260 Hz the signal completes only about one e-folding per ~1 cycle, i.e. it dies in 2–3 visible oscillations — a vivid illustration of the black hole's miserable quality factor and of why extracting even a single clean mode from real data is a genuine achievement.

Black-hole spectroscopy and the no-hair test

Here is why quasinormal modes are so prized. The no-hair theorem says a stationary black hole in general relativity is completely specified by mass, spin, and (astrophysically negligible) charge. So every quasinormal frequency, real and imaginary, is a known function of just M and a*. Detect the dominant mode and you solve for M and a*. Detect a second mode — an overtone (n = 1) of the same harmonic, or a subdominant angular harmonic like ℓ = 3, m = 3 — and general relativity makes a sharp, parameter-free prediction for its frequency and damping. If observation disagrees, the object is not a Kerr black hole.

Mode (ℓ, m, n)	Mω_R (Schwarzschild)	Mω_I	Q	Role
2, 2, 0 (fundamental)	0.3737	0.0890	2.10	Dominant tone
2, 2, 1 (first overtone)	0.3467	0.2739	0.63	Decays ~3× faster; spectroscopy target
3, 3, 0	0.5994	0.0927	3.23	Higher harmonic; tests no-hair
4, 4, 0	0.8092	0.0942	4.29	Weakly excited higher harmonic
2, 1, 0	0.3736	0.0890	2.10	Degenerate with 2,2 at a*=0; splits with spin

This is precisely analogous to atomic spectroscopy: hydrogen is identified not by one line but by the ratios of its lines, which the Rydberg formula fixes. A Kerr black hole is identified by the consistent ratios of its quasinormal lines. The strategy was formalised by Dreyer and collaborators (2004) and by Berti, Cardoso and Will (2006), who showed LISA could resolve many modes from supermassive-hole mergers and turn the no-hair theorem into a quantitative measurement.

The hidden link to the photon sphere

Why should the ringing frequency be what it is? In the eikonal (large-ℓ) limit there is a beautiful answer: the real part of the quasinormal frequency equals the orbital angular velocity of the unstable circular photon orbit, and the imaginary part equals the rate (the Lyapunov exponent) at which nearby photon orbits peel away from it. The ringdown is, in effect, a wave momentarily trapped just outside the light ring at r = 3GM/c² (for Schwarzschild): it circulates at the photon-sphere frequency — setting the pitch — and leaks out a little each orbit because the orbit is unstable — setting the damping.

ω_R ≈ ℓ × Ω_photon,    Ω_photon = c³ / (3√3 · GM)
ω_I ≈ (n + ½) × λ_Lyapunov

This correspondence is exact only as ℓ → ∞, but it is already accurate to ~10% for the ℓ=2 fundamental, and it explains the otherwise mysterious closeness of Mω_R ≈ 0.3737 to the photon-ring orbital value 1/(3√3) ≈ 0.1925 per unit ℓ. It also ties quasinormal modes to the bright photon ring imaged by the Event Horizon Telescope: same piece of spacetime geometry, two different probes.

Variants and related ringing

• Overtones (n > 0). Beyond the fundamental, each harmonic has an infinite tower of increasingly damped overtones. Recent work (Giesler et al. 2019) argued that including several overtones lets the ringdown model be pushed back almost to the merger peak, sharpening mass and spin recovery — though the physical reality of fitting many fast-decaying overtones to noisy data is debated.
• Kerr spin splitting. Rotation lifts the m-degeneracy: prograde (m > 0) modes rise in frequency and grow long-lived as a* → 1, while retrograde modes do the opposite. At the extremal limit a* = 1 the fundamental damping ω_I → 0 — the bell would, in principle, ring forever.
• Charged (Reissner-Nordström) and higher-dimensional holes. Adding charge or extra dimensions changes the potential and shifts the spectrum; these are theory laboratories rather than astrophysical cases.
• Exotic compact objects. Boson stars, gravastars and wormholes would ring with different spectra — and may emit late "echoes" because they lack a true absorbing horizon to drain the energy. A detected echo would be a smoking gun for new physics.
• Neutron-star and fluid analogues. Relativistic stars have their own quasinormal spectra (the f-, p-, and w-modes), and laboratory "analogue" systems — draining bathtub vortices, optical fibres — have demonstrated quasinormal ringing as a generic feature of leaky resonators.

Common misconceptions and subtleties

• "The black hole's surface vibrates." Nothing material oscillates. What rings is the curvature of spacetime in the strong-field region just outside the horizon; the "tone" is encoded in the emitted gravitational waves, not in any membrane.
• "Quasinormal modes are a complete basis." Unlike true normal modes, the quasinormal set is not complete and the modes are not orthogonal — the early ringdown also contains a prompt scattered burst, and the very late signal is a power-law "tail" (Price's tail, t^(−(2ℓ+3))) that no finite sum of modes reproduces. The clean exponential ringing is only an intermediate-time phenomenon.
• "More overtones always means a better fit." Overtones decay fast and are easily mimicked by noise; over-fitting can manufacture spurious agreement. Robust no-hair tests demand a genuinely high signal-to-noise event, which is why GW150914 (SNR ≈ 24) gave only marginal access to a second mode.
• "The damping is the energy crossing the horizon." Both channels matter: energy radiates to infinity as gravitational waves and is absorbed at the horizon. For the fundamental ℓ=2 mode, the bulk of the damping is radiation to infinity, with horizon absorption a smaller (but essential) contribution.
• "You can read mass and spin from the pitch alone." One real frequency is one number against two unknowns (M and a*). You need the damping time as well — or a second mode — to break the degeneracy. That is why ringdown analyses always fit the complex frequency, not just the audible pitch.