A spin that is measured as an angular momentum

Three routes to a black hole’s spin measure it by watching what the hole does to something else. Continuum fitting and the iron line both read where the accretion disc stops; Sołtan’s argument reads how much light a population of holes emitted per kilogram swallowed; an image reads the size of a capture cross-section for photons. All three infer an angular momentum from something that is not an angular momentum.

A merging pair does not require the intermediary. Two holes spiralling together radiate a waveform whose phase depends directly on their spins, through the same spin–orbit coupling that advances Mercury’s pericentre — no disc, no photons, no plasma, no radiative efficiency. It is the one measurement of black-hole spin with nothing in it but the metric.

What it returns is not what one would want.

What the phase actually depends on #

An inspiral is a slow sweep in frequency, and everything about the source is written in how fast that sweep runs. The combination of the two masses that sets the leading-order rate is the chirp mass, and it is measured to a fraction of a per cent because it enters first and the signal contains hundreds of cycles.

The spins enter three orders later, at what is called one-and-a-half post-Newtonian order, and they enter through a particular combination. The leading spin term in the phase is

$$\chi_{\rm eff} \;=\; \frac{m_1\chi_1\cos\theta_1 + m_2\chi_2\cos\theta_2}{m_1+m_2},$$

the mass-weighted average of the two spins’ components along the orbital angular momentum. The components perpendicular to it do not appear at this order, and neither does anything else about either spin.

The physical content of the combination is worth having, because it is not an artefact of an expansion. A spin aligned with the orbit exerts a spin–orbit torque that resists the inspiral: the pair hangs up, takes more cycles to reach a given separation, and merges at a smaller radius and a higher frequency. An anti-aligned spin does the reverse. So what the waveform is sensitive to is the total spin–orbit coupling, which is a sum weighted by mass and projected onto the orbital axis — and that is exactly the quantity written above.

The measurement is of a torque, not of a rotation. Two holes spinning rapidly in opposite directions exert no net torque and are indistinguishable, in the phase, from two that are not spinning at all.

The hang-up, in cycles #

It is worth putting a number on the effect the measurement rests on, because it is larger than a one-and-a-half-order term sounds.

A pair of thirty-solar-mass holes with no spin sweeps through the band of a ground-based detector in about two-tenths of a second, accumulating of order ten cycles above twenty hertz. Give both holes a spin of 0.8 aligned with the orbit and the spin–orbit torque holds the orbit up: the last stable orbit moves inward, the pair reaches a higher frequency before it merges, and the number of cycles in band rises by several. Anti-align the same spins and the opposite happens.

Several cycles out of ten is not a subtlety. It is a phase difference of many radians, and a matched filter is sensitive to a phase difference of a fraction of one — which is why a quantity entering at third order in a small parameter is nonetheless measured to a tenth.

The same arithmetic explains why it is the only spin quantity measured well. Precession does not change the number of cycles; it modulates their amplitude, and an amplitude is measured far less precisely than a phase because it is degenerate with the distance and the orientation. The subject’s whole hierarchy of well- and badly-measured parameters follows from that one distinction, and it is the same reason a merger’s distance is so much worse determined than its masses.

Why the degeneracy is not broken by more data #

The obvious response is that the perpendicular components must show up somewhere, and they do: they torque the orbital plane itself, so the whole orbit precesses about the total angular momentum, and the direction the radiation is beamed swings round a cone.

That modulation is the only thing that separates the individual spins, and it has a problem that no amount of observing removes.

The quantity that would break the degeneracy is suppressed by exactly the geometry that makes a merger detectable. That is not a coincidence in any deep sense — both follow from the same beaming pattern — but it is a genuinely unhelpful arrangement, and it is why a catalogue of a hundred events contains a few whose precession is measured and the rest whose individual spins are barely constrained.

There is a second, subtler difficulty. The effective spin is partially degenerate with the mass ratio, because both affect the rate of the frequency sweep in similar ways over the band an instrument actually hears. A pair with a low mass ratio and a positive effective spin produces nearly the same phasing as a more equal pair with a smaller one, and the posterior is a banana rather than an ellipse.

What the population says #

Because no single event’s spins are well measured, every conclusion about black-hole spin from this method is a statement about a distribution. That sounds like a weakness and is the opposite: the distribution of one well-measured number over a hundred events is a much stronger constraint than a hundred badly measured pairs of numbers would be.

The distribution’s mean sits a little above zero — of order 0.05 — and it has a tail on the negative side that is not consistent with zero. Both halves of that sentence are doing work.

A binary made from two stars that lived together inherits the plane they orbited in. Tides and mass transfer align the spins further, and the supernovae that made the holes can tilt them but not usually by much. Such a population has no negative tail at all, and a mean set by how fast the holes happen to be spinning.

A binary assembled by encounters in the core of a globular cluster has no such memory. Two holes that met by chance have spin directions drawn from nothing, so the tilts are isotropic, the distribution of the effective spin is symmetric about zero, and half the mergers are retrograde.

The observed distribution is neither. Its mean is positive and small, and its negative fraction is a few tens of per cent. The only thing that produces both is a mixture, and the quantity a population analysis actually reports is the mixing fraction — which is a statement about where black holes pair up, obtained from a number that has nothing to do with location.

Where the spins come from in the first place #

The distribution also says something about how fast the holes were turning before they met, and here the answer has been a surprise in the direction nobody planned for.

The individual spin magnitudes, inferred from the population rather than from single events, are small. Most of the holes that merge appear to be spinning at a few tenths of the maximum or less, and a population in which every hole was near the extremal limit is firmly excluded — a conclusion that sits awkwardly beside the spectral spins of accreting holes in X-ray binaries, which cluster above 0.9.

The two samples are not the same objects and the disagreement may be entirely about selection. A hole whose disc can be fitted is a hole that is accreting brightly, and prolonged accretion in a fixed plane is what spins a hole up; a hole that merges has usually not accreted much of anything. But the two methods also fail differently, and neither can check the other, for a reason worth stating on its own.

The spectral methods measure a magnitude and are blind to direction. The last stable orbit depends on the spin’s size and on whether the disc goes round with it or against it, and nothing else; a hole’s spin axis relative to anything outside the disc does not enter.

The waveform measures a direction-weighted sum and is nearly blind to magnitude. Two large spins pointing opposite ways return the same number as two small ones.

So the two families are complementary in precisely the sense that means they cannot be compared. There is no quantity both of them measure, and a disagreement between the samples cannot be attributed to either technique being wrong, because neither is measuring what the other is.

What the merger leaves behind #

There is a spin in this problem that is measured well, and it is the one nobody chose: the remnant’s.

The final hole’s spin is fixed almost entirely by the orbital angular momentum the pair carried at the last stable orbit, and that is a number the metric supplies. Two non-spinning holes of equal mass merge into a remnant at about 0.69 — not because either was rotating but because the orbit was, and the orbit’s angular momentum had nowhere else to go. For unequal masses the answer falls, because the reduced mass carries less; for aligned component spins it rises, and for anti-aligned it falls further.

That number is read directly off the ringdown, the damped oscillation the remnant settles through, whose frequency and decay time depend on the mass and the spin and nothing else. It is the closest thing this subject has to a spin measurement with no population argument, no accretion model and no projection in it.

The catch is that the remnant’s spin says nothing about the components’. A measured 0.67 is what almost any pair of slowly spinning holes produces, so the ringdown confirms the metric and adds nothing to the question of what the components were doing. The best-measured spin in the problem is the least informative one, which is what happens when a quantity is dominated by a term that carries no information about the initial state.

It does bear on one thing. A remnant near 0.7 is a plausible ingredient for a second merger, and a hole with a spin near 0.7 that shows up in a catalogue whose other members are near zero is a candidate for having been made that way. A handful of events have component spins high enough to raise the question, and it is currently the best evidence that holes merge more than once.

The events that carry the argument #

A statement about a distribution can be made without any individual event being decisive, and for most of this catalogue that is the situation. A few events are worth naming, because each of them is where one of the claims above actually bites.

The first confidently non-zero effective spin came from a pair of about fourteen and eight solar masses, whose posterior excluded zero at high confidence and whose value was around 0.2. It settled a question the first detection had left open — whether the holes that merge are rotating at all — and it settled it with a number small enough to be consistent with almost any formation story.

The unequal-mass events are where individual spins begin to separate. A pair with a mass ratio near a quarter carries higher harmonics in its waveform as well as the dominant one, and the extra harmonics break the degeneracy between the mass ratio and the effective spin that the last figure above is about. Those events give the tightest constraint on a primary hole’s spin in the catalogue, and the constraints are mostly upper limits.

Precession has been claimed for a handful of events and firmly established for none. The strongest case rests on an analysis whose result depends on how a detector artefact overlapping the signal was removed, and independent reanalyses have reached different conclusions — which is exactly the situation a marginal peak in a search always produces, and the right response is the one the field took: to report the disagreement rather than to pick a side.

And the most massive events are where the question of a second merger arises. A remnant of a first merger carries a spin near 0.7 whatever its parents did, so a component hole with a high spin and a mass in the gap where stellar collapse is not expected to leave anything is a candidate for having been built rather than born. It is currently a handful of objects and a plausible story, not a measurement.

What was actually measured #

A strain time series from two or three interferometers, of duration between a fraction of a second and a couple of minutes, matched against a bank of modelled waveforms.

Every number quoted here is a parameter of that model. The spins are not measured against anything external; they are the values that make a computed waveform agree with the data, and their errors are the width of a posterior in a fifteen-dimensional space that also contains the two masses, the distance, two sky angles, an inclination, a polarisation, a phase and a time.

Two consequences follow, and both are structural.

The model must be right. A waveform family that mis-modelled the spin terms by a small amount would return spins that were wrong by a large one, and the defence is that several independent families — built from post-Newtonian expansions, from numerical relativity, and from calibrated hybrids — are compared against one another on every event. They agree on the effective spin far better than they agree on the individual components, which is the same ordering the data impose.

And the population is a selected one. What a survey could have seen is not what it did see: a positive effective spin delays the merger and raises the frequency at which it happens, which changes where in the band the signal sits and therefore how loud it is. The correction is not large and it is not zero, and every population statement above has it folded in.

Where the model stops #

A waveform is not an observation of a hole. Everything here is a parameter of a template, and a template is a solution of the field equations for two point masses with spins. The assumption that the objects are black holes, rather than something else compact enough to reach the same frequencies, is built into every template in the bank and is not tested by fitting one.

The tilts are quoted at a reference frequency. A precessing binary’s spin directions change through the inspiral, so “the tilt” has to be defined somewhere, and the convention — usually twenty hertz, or an epoch a fixed time before merger — is a choice. Comparisons between analyses that chose differently are comparisons of different quantities.

The spin magnitudes come from a prior. Nothing in most events’ data prefers one spin magnitude over another, so the posterior on an individual spin is close to the prior that was imposed, and the standard prior is uniform in magnitude and isotropic in direction. A population inference that treats those posteriors as measurements is inferring partly from the prior, and the modern analyses handle it by fitting the population and the events together rather than in sequence.

And the sample is small and shallow. A hundred events, drawn from a volume that reaches a few billion light years for the loudest and a few hundred million for the quietest, is not a census of anything. The pairs that merge are also the pairs whose orbits had already circularised, which is a selection on the assembly history as surely as the loudness is a selection on the orientation.

Still open: what an individual spin would take #

What comes next is a measurement of a component spin rather than of a combination, and the requirement is stated in the second figure of this essay: a signal seen from the side, with enough cycles in band to resolve several precession periods. For stellar-mass holes that means a detector sensitive below ten hertz, which no ground-based instrument can be — the ground itself moves too much.

Beside it sits the pair of quantities treated as one here. The in-plane spin is summarised by a single effective number in the same way the aligned components are, and it carries the same kind of degeneracy; whether the two in-plane components can be separated at all, for any source, is an open question with a clean answer in principle and no instrument in practice.

And under both, the thing all of this is really about. A measurement of a projection is not a measurement of a vector, and the projection is what the physics couples to. Every method recovers some particular functional of a hole’s angular momentum — an orbital radius, a radiative efficiency, a capture cross-section, a torque — and none of them recovers the angular momentum. The methods disagree about black-hole spin, and the first thing to establish about any such disagreement is whether the two sides are functions of the same argument.