A few cycles that are only mass and spin

The calculation of what an orbit radiates follows two masses in orbit as the energy they radiate shrinks the orbit and speeds them up, in a runaway that takes ten to the twenty-three years for the Earth and the Sun and eight minutes for the last thousand kilometres of a pair of black holes.

That calculation ends where its assumptions do. It treats the two bodies as points in a nearly flat spacetime moving on a nearly Newtonian orbit, and by the end of an inspiral none of those holds. What comes after is not another orbit. It is one object — and its mass is less than the sum of what went in, by the energy the waves carried off.

That object is distorted — it was made by two things falling together — and a distorted object settles down. How it does so is the subject here, and the answer is unlike any other relaxation in this collection, because a black hole has nothing to relax into except a shape that is completely specified by two numbers.

A resonance with a complex frequency #

An ordinary resonance has a frequency and a width, and the width is a lifetime: a mode that loses energy cannot have a sharp frequency, and the sharper the resonance the longer it rings. Writing the two as one complex number — a real part that is the pitch and an imaginary part that is the decay rate — is a convenience everywhere else and is closer to necessary here.

The reason it is necessary is that a black hole’s modes are not normal modes in the usual sense. A normal mode of a drum is a standing pattern that would persist forever if nothing damped it, and the damping is an addition. A black hole’s modes cannot be separated that way, because energy leaves them in two directions at once and neither leak is optional: outward to infinity as radiation, and inward through the horizon, which absorbs everything and returns nothing.

So there is no undamped version of the problem to perturb. The modes are quasinormal, they have complex frequencies from the start, and the boundary conditions that select them — purely outgoing at infinity, purely ingoing at the horizon — are what makes the problem non-standard mathematically. They are the reason the spectrum is not the eigenvalue problem of a self-adjoint operator and does not come with the guarantees one of those carries.

Two numbers, and what follows from there being only two #

The no-hair theorems say that a stationary black hole in general relativity is completely described by its mass, its angular momentum and its electric charge — and astrophysical holes have no charge worth mentioning, because any they acquired would be neutralised by the surrounding plasma within moments.

That is a statement about the object. Its consequence for a measurement is sharper: every quasinormal frequency and every damping time is a function of two numbers. Not approximately, and not for the lowest few modes — for all of them, exactly, within general relativity.

The numbers in these figures come from fits to the numerically computed spectrum — the standard ones, of the form $M\omega_R = f_1 + f_2(1-a)^{f_3}$ with three coefficients per mode — which reproduce the computation to about a per cent across the range of spin drawn. They are fits, and the check on them is the one case that is known exactly: at zero spin the fundamental must come out at $0.3737 - 0.0890i$ in units of the inverse mass, and it does.

For a remnant of 62 solar masses that is 192 hertz and a damping time of 3.5 milliseconds. Spin it up to 0.68 and the pitch rises to 274 hertz and the damping time to 3.7 milliseconds. Both numbers are in the band a ground-based interferometer hears best, which is not a coincidence — the band was chosen for the mergers of exactly these masses — and both were measured for the first detected event, at the far end of a response that falls away above and below.

The pitch is a photon orbit, and the damping is how it comes apart #

The frequencies are computed numerically and they have a picture, and the picture is exact enough to be useful.

Outside a black hole there is one radius at which light can travel in a circle. For a non-spinning hole it is at one and a half times the horizon radius — the circle light cannot leave and cannot stay on — and the orbit is unstable: a photon a little inside spirals in, a photon a little outside spirals away, and neither stays.

A wave bouncing around near that radius is trapped in the same unstable way. It circulates at the orbital frequency of the light ring, and it leaks away at a rate set by how fast neighbouring trajectories separate from that orbit. Both numbers are properties of the null geodesics and are computed from the geometry without solving any wave equation.

For a Schwarzschild hole the light ring’s orbital frequency is $c^3/3\sqrt3\,GM$, so a wave with angular index $\ell$ going round it would have $M\omega_R \approx \ell/3\sqrt3$. For $\ell = 2$ that is 0.385 against the computed 0.3737 — three per cent. The rate at which nearby orbits diverge, the Lyapunov exponent of the unstable orbit, is the same $c^3/3\sqrt3\,GM$, and the fundamental’s damping rate should be half of it: 0.0962 against the computed 0.0890, eight per cent.

A picture that gets both parts of a complex frequency to within a few per cent, from a geodesic calculation with no wave in it, is not a coincidence. It is the short-wavelength limit of the wave equation, and it becomes exact as $\ell$ grows: the agreement is three per cent at $\ell = 2$ and better than one at $\ell = 4$.

What it explains is why the spectrum is so rigid. The light ring’s position and the divergence rate there are fixed by the mass and the spin because the geometry is, and there is nothing else in the neighbourhood for a mode to depend on. It also explains the extremal limit: as the spin approaches its maximum the prograde light ring approaches the horizon, the divergence rate goes to zero, and the modes stop damping.

And it gives a reason to expect the whole scheme to be robust. The frequencies are a property of a region well outside the horizon — one and a half horizon radii for a non-spinning hole — so a ringdown measurement is not a probe of whatever is or is not at the horizon itself. That is a limitation and a reassurance at once: it is why an object with a surface just outside where a horizon would be can ring almost identically at first, and why the interesting deviations show up not in the modes but in what arrives afterwards.

Why one cycle is the whole of it #

The quality factor of the dominant mode is about three at a spin of 0.68 and two at zero spin, which means the ring lasts about one cycle.

That is worth dwelling on, because the word “ringdown” invites a comparison with a bell and the comparison fails in every particular. A struck bell sustains for thousands of cycles; its pitch is sharp enough to tune an orchestra by; and which of its modes are excited depends visibly on where it was struck. A black hole’s fundamental mode is gone within a cycle or two, its “pitch” has a width comparable to itself — a resonance so broad it barely counts as one — and extracting even one frequency from the data requires knowing where the ringdown began to within a fraction of a cycle.

The reason is the horizon. A bell rings for a long time because it loses energy slowly — a little to the air, a little to the mount. A perturbed black hole loses energy at the fastest rate the geometry allows, because nothing that crosses the horizon comes back and the mode has one foot inside it. The damping is not a loss mechanism added to an oscillator; it is half of what the oscillator is.

The consequence for measurement is severe and is the central practical difficulty of the subject. A signal with a quality factor of three contains very little information: the number of independent numbers that can be extracted from a damped sinusoid of $N$ cycles buried in noise grows with $N$, and $N$ here is one. Detecting a second mode, which is what the test in the next section requires, needs a signal-to-noise ratio in the ringdown alone of order tens, and the loudest events so far give a few.

The test that a second mode makes possible #

Here is the structure of the argument, and it is unusually clean.

Measure one frequency and one damping time and there are two equations in two unknowns; the mass and the spin follow. Nothing has been tested at that stage — two measurements have determined two parameters, which is what measurements do.

Measure a second mode and there are four equations in the same two unknowns. The theory predicts the second mode’s frequency and damping time from the mass and spin the first mode gave, with no freedom left anywhere, so the two extra numbers are predictions that can fail.

They would fail if the remnant had any property beyond mass and spin. A surface instead of a horizon, an exotic interior, a modification to gravity at the horizon scale, an extra field the hole carries — anything at all beyond the two numbers the area theorem also depends on — every one of those adds a parameter, and a spectrum with an extra parameter in it does not have to put four curves through one point.

That is the whole of black-hole spectroscopy: the first mode measures and every mode after it tests, and the reason it is a clean test rather than a consistency check is the no-hair theorem’s strictness. There is no “roughly two parameters”. Either the spectrum is a function of two numbers or the object is not what general relativity says it is.

What has actually been measured #

The observational record is short and worth stating precisely, because the gap between what the argument needs and what exists is the subject’s whole present state.

The first detected event, in 2015, had a remnant of about 62 solar masses and a total signal-to-noise ratio of 24. Of that, the part after the peak — the part a ringdown analysis may use — was a few. The fundamental frequency and damping time were consistent with the mass and spin inferred from the inspiral, which is a consistency check between two halves of one signal rather than the four-curve test above, and it is the strongest statement the data supported.

The heaviest event so far, in 2019, produced a remnant of about 142 solar masses. A heavier remnant rings at a lower frequency, which for a ground-based detector means further into the band where the noise rises steeply, and it also means fewer inspiral cycles — so the mass and spin came largely from the merger and ringdown themselves rather than from an independent inspiral measurement. That is the opposite trade from the first event and it is why the two are complementary rather than cumulative.

Across all detections the position is the same: the ringdown is seen, the fundamental mode is measured with fractional uncertainties of tens of per cent, and a second mode has not been established beyond dispute. Tens of per cent is enough to exclude gross departures and is very far from the per cent or better a genuine no-hair test requires.

The arithmetic of what would be needed is straightforward. The uncertainty on a frequency extracted from a damped sinusoid falls roughly as the inverse of the signal-to-noise ratio in the ringdown, so a per-cent measurement of two modes needs a ringdown signal-to-noise of order a hundred against the few available now. That is a factor of tens, and since the strain scales as the inverse of the distance, it is not obtainable by waiting for a nearer event of the same kind — it is obtainable by building a quieter instrument or by observing heavier holes from space.

The spectrum is predicted and the amplitudes are not #

The spectrum is predicted and the amplitudes are not. How loudly each mode is excited depends on how the remnant was made — the mass ratio, the two spins and their orientations — and comes from numerical simulations of the merger rather than from the theory of the final object. The amplitudes in the first figure are the rough values a merger of comparable masses produces and are illustrative. That separation is a feature rather than a defect: the tested quantities are the ones the theory fixes.

When the ringdown starts is not defined. There is no instant at which the merger ends and a perturbed Kerr hole begins; the transition is gradual, and the fitted frequencies depend on where the fit is started. Starting too early includes nonlinear merger physics the mode description does not cover; starting too late throws away most of the signal-to-noise, since the amplitude is falling by a factor of e per damping time. Whether the overtones extracted from the earliest part of the data are genuine quasinormal modes or artefacts of fitting damped sinusoids to something else has been argued since 2019 and is not settled.

The fits are fits. The coefficients used here reproduce the computed spectrum to roughly a per cent, which is far better than any measurement and far worse than the exact result. Near extremal spin they degrade, and the extremal limit itself — where the damping goes to zero and the modes accumulate — is a singular case these expressions approach rather than describe.

And a real remnant is not stationary. It is recoiling, it may be surrounded by matter, and it is still absorbing whatever did not escape during the merger. All three perturb the spectrum, all three are small for the events observed so far, and all three would matter for a test at the precision the argument deserves.

The noise, the tail, and the wave that goes down the hole #

They cannot show the noise. Every curve here is an exact function, and the measurement it stands for is a fit to a few milliseconds of data in which the signal is comparable to the noise. The four curves in the consistency figure are drawn as lines and are in reality bands whose width is set by the measurement; whether they intersect is a statement about overlapping bands, not about crossing curves, and a test passes or fails according to how wide the bands are.

Nor can they show the modes that are not oscillations at all. The quasinormal spectrum is not the whole response: a perturbed black hole also produces a power-law tail, arising from backscattering off the spacetime curvature far from the hole, which persists long after every exponential has died. It is far too faint to detect and it is the part of the signal that formally never ends.

And they cannot show the horizon’s own contribution. A quasinormal mode is a property of the spacetime outside the hole, and the wave that goes down the horizon is as real as the one that comes out to a detector — it carries energy and angular momentum into the hole and is what makes the remnant’s final mass smaller than the sum of what went in. Nothing about that half of the radiation is observable from outside, and it is half of why the mode damps.

Still open: whether the overtones have been measured at all #

The first claim of a second mode in real data was made in 2019, using the fundamental’s first overtone in GW150914, and it was followed by analyses that found the evidence much weaker and by others that defended it. The dispute is about the start time: the overtone decays in about a millisecond, so the evidence for it lives in the first millisecond after the peak, which is exactly where the merger is still in progress and where the linear mode description has no right to hold.

What would settle it is a louder event. The signal-to-noise in the ringdown scales with the total mass for fixed distance, and a remnant of several hundred solar masses close enough would put the overtone question beyond argument — as would the next generation of ground-based detectors, which would collect ringdowns at a signal-to-noise of hundreds rather than tens. A space-based detector observing a merger of two million-solar-mass holes would resolve a dozen modes and turn the test in this essay from a consistency check into a precision measurement.

The habit worth carrying away concerns what makes a prediction testable. A theory earns a test when it fixes more observable numbers than it has free parameters, and the count is the whole of it. Two parameters and two measurements is a determination; two parameters and four measurements is a test; and the strictness of the no-hair theorems is exactly that they refuse to supply a third parameter when the fourth measurement disagrees.