Black Hole Spectroscopy from First Principles
Two black holes spiral together and collide. For a few milliseconds afterward, the single object they leave behind is deformed, and it sheds that deformation as gravitational waves: a tone that decays like a struck bell. One question: what can you learn about the bell from the recording? General relativity's answer is extreme. The entire tone list, every pitch and every decay rate, is fixed by exactly two numbers.
Waveform plots show strain against time. Strain is dimensionless and tiny (about 10⁻²¹ for the loudest event ever seen), so the vertical axes are in arbitrary units; only the shapes and ratios matter.
Time is shown two ways. Figures tied to a specific detection use milliseconds, assuming a GW150914-like remnant. Figures about black holes in general use the geometric time unit M, explained in Section 4, because every quantity in the problem scales with the black hole's mass. One M is about 0.33 ms for that same remnant, so the two clocks are easy to convert between.
1What happens when two black holes merge
The signal in Figure 1 has three phases, and they are physically different regimes, not just visual ones.
Inspiral
Two black holes orbit each other. Orbiting masses radiate gravitational waves, radiated energy comes out of the orbit, and the orbit shrinks. Smaller orbit means faster orbit means stronger radiation, so the process runs away: the wave's frequency and amplitude both climb. Played through a speaker, this rising sweep is the famous chirp.
Merger
The horizons touch and the two objects become one. This is the regime with no good pen-and-paper description; it is where numerical relativity, solving Einstein's equations on a supercomputer, is unavoidable. It is also when the signal is loudest.
Ringdown
The newly formed remnant is a single black hole, but a misshapen one. A deformed black hole cannot stay deformed: it radiates the deformation away and settles into its final, stationary shape. The radiation from this settling is the ringdown, and its mathematical form is astonishingly simple: a sum of damped sinusoids.
What a detector like LIGO actually records is strain: the fractional change in the length of its arms as the wave passes, h = ΔL/L. For GW150914, the first detection, the peak strain was about 10⁻²¹. Over a 4 km arm that is a length change of 4 × 10⁻¹⁸ m, roughly a thousandth of a proton diameter, which is why it took a hundred years to build an instrument that could see it.
The numbers for that first event set the scale for everything in this post. Two black holes of about 36 and 29 solar masses merged into a remnant of about 62; the missing 3 solar masses left as gravitational waves in a few tenths of a second. The remnant that did the ringing has a detector-frame mass of about 68 solar masses and a spin of 0.69, and those two numbers are the ones every figure below defaults to.
The key fact about the remnant: it is a Kerr black hole, the exact solution of Einstein's equations for a rotating black hole, and a Kerr black hole is completely described by its mass M and its spin. Not approximately described: completely. Every other detail of the messy merger, the masses of the parents, their spins, how they hit, is either radiated away or swallowed. This is the no-hair theorem, and the ringdown is the place in nature where it is most directly on display.
2The ringdown is a struck bell
Strike a bell and the sound you hear is not noise: it is a handful of discrete tones, each dying away at its own rate. The ringdown is exactly this, written in spacetime. Each tone is a quasinormal mode (QNM), and each one is a damped sinusoid:
A amplitude at t = 0 · φ starting phase · f frequency (the pitch) · τ damping time (how fast it dies)
Two of those four numbers, f and τ, are properties of the black hole itself; the other two just record how hard and at what moment the bell was struck. The damping time has a concrete meaning you can check on the plot below: after one τ the amplitude has fallen to 1/e ≈ 37% of where it started, after two it is at 13%, after three at 5%.
Why damped, and why is the mode only "quasi" normal? A violin string has true normal modes: the energy stays on the string, so an idealized string would ring forever. A black hole cannot hold on to its oscillation energy. The wave escapes in both directions at once, outward to infinity (that is the part we detect) and inward through the horizon, which absorbs perfectly and gives nothing back. An oscillator that leaks is described by a complex frequency,
real part: oscillation · imaginary part: decay. One complex number per tone.
and "quasinormal" is just the standard name for the modes of such an open, leaking system. The leak is severe. A useful figure of merit for any bell is the quality factor Q = π f τ, roughly how many oscillations happen before the sound dies. A wine glass has Q in the thousands. The dominant mode of a merger remnant has Q ≈ 3: the loudest sound in the universe rings about three times and is gone. Nearly every difficulty in the rest of this post traces back to that one number.
3The mode labels (ℓ, m, n)
A bell has many tones, and so does a black hole. Each quasinormal mode is labeled by three integers, and the labels answer two different questions.
(ℓ, m) — the shape on the sky
The angular pattern of the oscillation, exactly like the spherical-harmonic labels in the hydrogen atom. ℓ = m = 2 is the quadrupole: the pattern of a sphere being squeezed along one axis while bulging along another, then swapping. Two orbiting masses are a rotating quadrupole, so the merger pumps this mode hardest, and (2,2) dominates essentially every observed signal.
n — the overtone index
For each angular shape there is a whole ladder of modes with the same (ℓ, m), ordered by how fast they die. n = 0 is the fundamental, the longest-lived. n = 1 is the first overtone: nearly the same pitch, but dying roughly three times faster. Overtones matter early in the ringdown and are gone almost immediately.
So "the (2,2,0) mode" reads: quadrupole shape, fundamental tone. Here are the actual numbers for the three modes this post keeps returning to, at spin 0.69, with the conversions for a 68-solar-mass remnant:
| Mode | Name | f̂ | τ̂ (M) | f (Hz) | τ (ms) | Q = πfτ |
|---|---|---|---|---|---|---|
| (2,2,0) | fundamental | 0.0841 | 12.3 | 251 | 4.12 | 3.25 |
| (2,2,1) | first overtone | 0.0822 | 4.07 | 245 | 1.36 | 1.05 |
| (3,3,0) | higher harmonic | 0.1339 | 11.9 | 400 | 3.99 | 5.01 |
4Mass sets the clock, spin sets the shape
Where do the numbers in Figure 3 come from? General relativity has no free parameters here. For a Kerr black hole, every mode frequency is a known, tabulated function of the spin, scaled by the mass:
ω̂ℓmn(χ): dimensionless Kerr frequency, depends only on spin · Ms: the black hole's mass expressed as a time
Ms = GM/c³ deserves a pause, because it is the unit trick the whole field runs on. Take the mass, multiply by G, divide by c³: the result is a time, about 4.93 microseconds per solar mass. It is the natural tick of the black hole's clock, and writing every time in units of it (that is the "M" on the axes) makes every black hole identical up to scale. Double the mass and the ringdown has exactly half the frequency and exactly twice the damping time; the waveform is the same shape, stretched. The dimensionless spin χ = cJ/GM² ∈ [0, 1), the angular momentum measured against the maximum a black hole of that mass can carry, is the only knob left, and it bends the tone list into a different shape.
The figure below is the entire content of the no-hair theorem, drawn as a map. Each curve is where one mode is allowed to live, traced out as spin runs from 0 to 0.95. Pick a spin and each mode is pinned to a single point.
Now the payoff, and the reason this field is called spectroscopy. In the nineteenth century, chemists identified elements by their spectral lines: hydrogen's lines sit in fixed ratios, so two measured lines both pointing to the same element confirm it. The black-hole version:
- Measure one mode (a frequency and a damping time): two numbers in, two unknowns out. Solve for M and χ. No test yet, just a measurement.
- Measure a second mode: general relativity now predicts its frequency and damping time with zero free parameters, because M and χ are already spent. If the second mode lands where Kerr says it must, the no-hair theorem passed. If it does not, either the remnant is not a Kerr black hole or gravity is not general relativity.
That is the entire program. It needs two modes, measured well, from one event. And that turns out to be brutally hard, for reasons the next section makes quantitative.
5How a mode is identified, and when it can't be
Fitting, not listening
Nobody hears tones in the data. The detector output is the signal buried in instrument noise, and the analysis is a fit: propose a model, a sum of damped sinusoids with unknown (A, φ, f, τ) for each mode, and find the parameter values that best explain the stretch of data after the merger. The loudness of a signal relative to the noise is its signal-to-noise ratio (SNR, written ρ); GW150914's full signal had ρ ≈ 24, of which the ringdown portion carried only a fraction.
A fit returns best values and uncertainties, and the uncertainties are the whole game. The standard tool for forecasting them is the Fisher information matrix, which converts a signal shape and a noise level into the best possible error bars any unbiased analysis could achieve (the Cramér–Rao bound). Its most useful output is a scaling law: every uncertainty shrinks like 1/ρ. Twice the SNR, half the error bar. So for any yes/no question about resolving modes, there is a critical SNR above which the answer becomes yes.
The Rayleigh criterion
The classic yes/no rule is borrowed from optics. Two telescope images are "resolved" when their separation exceeds their blur; two modes are declared resolved when the separations between their parameters exceed the uncertainties on those parameters:
resolvable when both separations beat both blurs; each σ shrinks as 1/ρ, so this defines a critical SNR
For the fundamental–overtone pair, the naive version of this picture is doomed from the start, and Figure 6 shows why. A tone that lasts a time τ does not have a sharp frequency: its spectrum is a bump of width 1/(2πτ) (in optics this is a Lorentzian line; in quantum mechanics the same fact is the energy–time uncertainty relation). With Q ≈ 3, the bumps are enormously wide compared to the 2.2% frequency split.
The part the pretty picture hides: correlations
There is a second, sneakier problem. A fit's uncertainties are not private property of each mode. When two templates look alike, the fit can trade one mode's parameters against the other's while barely changing the summed waveform, and the error bars on everything inflate together. The bookkeeping for "how alike" is the overlap μ: the normalized inner product of the two mode templates, 0 for templates that share nothing, 1 for identical ones. From μ follow two standard diagnostics:
κ is the condition number of the 2-mode Gram matrix: how close the pair is to being one template in disguise
For long windows and nearly equal frequencies, the overlap has a closed form that is worth carrying around, because it exposes what actually controls the problem:
the ratio of the geometric to the arithmetic mean of the two damping times: the frequency split barely enters
Plug in the factor-of-three damping ratio from Figure 3 and you get μ ≈ 0.866, before the frequencies are even consulted. The criterion is named after frequency resolution, but for this pair the frequency split is a spectator. The price of the overlap is what my paper on this subject measures: with the correlations properly kept, the SNR needed to resolve the GW150914-like pair is a factor of about four higher than a correlation-blind estimate suggests, roughly 310 instead of 71. Play with both knobs below and watch which one actually moves μ.
Has it been done?
Once, maybe. In 2019, an analysis of GW150914 reported evidence for the (2,2,1) overtone alongside the fundamental, which would make it the first no-hair test with two modes from one event. In 2022 a reanalysis disputed the claim, finding the evidence depends delicately on exactly when the fit starts and how the noise is treated, for precisely the reasons above: the overtone is a fast-dying template nearly parallel to the fundamental, hunted at an SNR far below the thresholds of the previous paragraph. The disagreement is, at the time of writing, unresolved. Next-generation detectors, with ringdown SNRs in the hundreds, are expected to settle the question the boring way: with signals loud enough that the error bars stop being the story.
6The glossary
Every term this field expects you to know, in the order you met them.
gravitational wave
A ripple in spacetime itself, radiated by accelerating masses, travelling at the speed of light.
strain, h
What a detector measures: the fractional length change of its arms, h = ΔL/L. Dimensionless and around 10⁻²¹ for loud events.
inspiral / merger / ringdown
The three phases of a binary black hole signal: the accelerating orbit (the chirp), the collision itself, and the settling of the remnant.
remnant
The single black hole left after the merger. The subject of everything in this post.
Kerr black hole
The exact rotating-black-hole solution of Einstein's equations. Fully described by mass and spin.
no-hair theorem
The statement that an isolated black hole is completely characterized by mass and spin: no other memory of its history survives.
quasinormal mode (QNM)
One tone of a rung black hole: a damped sinusoid with frequency f and damping time τ, equivalently one complex frequency ω = 2πf − i/τ. "Quasi" because the system leaks energy and the tones decay.
(ℓ, m)
The angular-shape labels of a mode, like spherical harmonics. The quadrupole ℓ = m = 2 dominates mergers.
overtone index, n
Orders the modes of one angular shape by damping: n = 0 is the long-lived fundamental, n = 1 the first overtone, dying about 3× faster at nearly the same pitch.
damping time, τ
Time for a mode's amplitude to fall to 1/e ≈ 37%. Sets the spectral line's width, 1/(2πτ).
quality factor, Q
Q = πfτ: how many times the bell rings before dying. Black holes: about 3. Wine glass: thousands.
geometric time unit, M
Ms = GM/c³ ≈ 4.93 μs per solar mass: the black hole's natural clock tick. In these units all black holes of a given spin are identical.
dimensionless spin, χ
χ = cJ/GM² ∈ [0,1): angular momentum as a fraction of the maximum allowed. The one knob that changes the dimensionless tone list.
black hole spectroscopy
Measuring two or more QNMs from one event and checking they point to the same (M, χ): the observational test of the no-hair theorem.
signal-to-noise ratio (SNR, ρ)
The loudness of a signal relative to detector noise. Every parameter uncertainty scales as 1/ρ.
template
The parametrized model waveform being fit to data; here, a damped sinusoid per mode with unknown (A, φ, f, τ).
Fisher matrix / Cramér–Rao bound
The standard forecast machinery: from signal shape and noise level to the smallest error bars any unbiased fit can achieve.
Rayleigh criterion
The resolvability rule: two modes count as distinguished when their frequency and damping-time separations both exceed the uncertainties.
overlap, μ
Normalized inner product of two mode templates: 0 = orthogonal, 1 = identical. For the contested pair, about 0.86, set almost entirely by the damping-time ratio.
condition number, κ
κ = (1+μ)/(1−μ) for two templates: how close the pair is to degenerate. Near 14 for the contested pair; correlated errors inflate accordingly.
7Practice problems
Ten problems, all doable on paper in a few minutes each. Work them before opening the solutions; the arithmetic is the point, because these are exactly the conversions you do constantly when reading ringdown papers.
A · Units and bells
Convert the fundamental mode of the table in Figure 3 to physical units for a 68-solar-mass remnant: find f in Hz and τ in ms.
Solution
f = f̂ / Ms = 0.0841 / 3.35×10⁻⁴ ≈ 251 Hz
τ = τ̂ · Ms = 12.3 × 3.35×10⁻⁴ ≈ 4.1 ms f ≈ 251 Hz, τ ≈ 4.1 ms. Both match the measured GW150914 ringdown, which is why "250 Hz and 4 ms" is the most quoted pair of numbers in the field.
Compute the quality factor of that mode, and the number of full oscillation cycles completed in one damping time.
Solution
cycles in one τ = f · τ = 251 × 0.00412 ≈ 1.03 Q ≈ 3.2; about one cycle per e-fold of decay. By three cycles the amplitude is down 95%. This is why raw ringdown data looks like two or three visible wiggles, not a tone.
LISA, the planned space detector, targets remnants of around a million solar masses. Where does the fundamental mode land for M = 10⁶ M☉, χ = 0.69?
Solution
f = 0.0841 / 4.93 ≈ 0.017 Hz = 17 mHz
τ = 12.3 × 4.93 ≈ 61 s 17 mHz and a one-minute ringdown. Millihertz is hopeless on the ground (seismic noise) and exactly where a space interferometer is quietest: same physics, five orders of magnitude apart, pure mass scaling.
The fit starts 5 M after the peak instead of at the peak. What fraction of each mode's amplitude survives to the start of the fit?
Solution
overtone: e^(−5/4.07) = e^(−1.229) ≈ 0.29 67% of the fundamental, 29% of the overtone. Five mass units of patience costs you two-thirds of the overtone. Hence the fight over fit start times: the overtone only exists at the beginning, which is also where the "sum of damped sinusoids" model is least trustworthy.
B · The mode table
From the Figure 3 table, compute the fractional frequency split and the damping-time ratio of the (2,2,0)–(2,2,1) pair.
Solution
τ₀/τ₁ = 12.3/4.07 ≈ 3.0 A 2.3% split in pitch; a factor 3.0 in decay. The pair is nearly a unison in frequency and nothing alike in damping: remember which of the two carries the information.
Compute each mode's spectral half-width in f̂ units, 1/(2πτ̂), and compare with the frequency split between them.
Solution
width₂₂₁ = 1/(2π·4.07) ≈ 0.0391
split / width₂₂₀ = 0.0019/0.0129 ≈ 0.15 The split is 15% of the narrower line's width. In an optics textbook these two lines are one line. Any resolving that happens here is resolving by damping time, whatever the criterion's name suggests.
A ringdown is measured: f = 175 Hz, τ = 5.9 ms. Assuming it is the (2,2,0) mode at χ = 0.69, recover the remnant mass, then verify the damping time is consistent.
Solution
M = 4.81×10⁻⁴ / 4.93×10⁻⁶ ≈ 98 M☉
check: τ = 12.3 × 4.81×10⁻⁴ ≈ 5.9 ms ✓ M ≈ 98 M☉, and the damping time checks. This is the first half of spectroscopy: one mode spends the two unknowns. A second mode would now be a zero-parameter prediction.
C · Resolvability and correlations
Using the equal-frequency closed form, compute the overlap μ for a damping ratio of exactly 3.
Solution
From that overlap, compute the condition number κ and the amplitude-error inflation factor (1−μ²)^(−1/2).
Solution
(1−μ²)^(−1/2) = (1−0.75)^(−1/2) = (0.25)^(−1/2) = 2.0 κ ≈ 14; every amplitude error bar doubles. And this is only the amplitude-level penalty: letting frequencies and damping times float too, the full joint fit inflates the required SNR by about 4× for this pair.
A correlation-blind forecast says the pair is resolvable at ρ = 71; the full joint-fit calculation says ρ = 310. Uncertainties scale as 1/ρ. If an event delivers ringdown SNR 30, how far is each forecast from declaring the pair resolved?
Solution
joint: needs 310/30 ≈ 10× more SNR A factor 2.4 versus a factor 10. Since SNR grows with detector sensitivity, the two forecasts disagree by years to decades on when the no-hair test becomes routine, which is why the bookkeeping of correlations is not pedantry.