The second number a black hole has

A black hole is the simplest macroscopic object there is. Whatever fell in — hydrogen, iron, a star, a magnetic field, a library — the result is described by a mass, an angular momentum, and a charge, and the charge of any astrophysical hole is zero because the universe is full of free charges that would neutralise it in moments. Two numbers, then.

The collection has spent a good deal of effort on the first. Weighing the object at the centre of the galaxy took a complete stellar orbit and Kepler’s third law; an inspiral’s frequency sweep delivers a mass combination for a merging pair; a flare’s rise time bounds one from above. The second number is harder to measure and does more than it looks as though it should.

Why there is a last stable orbit at all #

In Newtonian gravity there is no innermost orbit. A circular orbit exists at every radius; it is stable at every radius; and a particle with any angular momentum whatever is kept away from the centre by the wall that angular momentum builds in the effective potential.

General relativity adds a term. The effective potential acquires a piece going as $-GML^2/c^2r^3$, which is attractive and steeper than the centrifugal barrier is repulsive, so at small enough radius the barrier is overwhelmed. The consequence is that the potential’s minimum and maximum — the stable circular orbit and the unstable one — approach each other as the angular momentum is reduced, and at a critical value they merge and vanish. For a non-rotating hole the merge happens at exactly six gravitational radii. That number does not depend on the hole’s mass, because a gravitational radius already has the mass in it: a hole of ten solar masses has its last stable orbit at ninety kilometres and one of four million has it at thirty-six million, and both are at six $GM/c^2$.

What the spin does to it #

A rotating mass drags the geometry around with it. Not the matter — the geometry: a gyroscope at rest with respect to distant stars precesses, and a particle with zero angular momentum released from far away acquires a rotation in the hole’s sense as it falls.

For an orbit going the same way as the hole, the dragged frame is already moving in the direction of motion, and the orbit can hold together closer in. For an orbit going the other way, the drag opposes it and the orbit must stay further out. At the extremal spin the two limits are one and nine gravitational radii, and the prograde limit sitting exactly at the horizon radius is not a coincidence — it is the same degenerate limit approached from two directions.

The consequence for energy is direct. A circular orbit at radius $r$ has a specific energy less than its rest energy, and the deficit is what has been radiated on the way in. The deeper the last stable orbit, the larger that deficit.

That is the number that makes quasars possible. A disc has to throw angular momentum away before anything can fall in, and the light it emits doing so is set by how deep the material gets before it stops orbiting. At 5.7 per cent a solar mass a year lights a galaxy; at 30 per cent it does so five times over.

The measurement, and why it is contested #

The spin is read from the shape of a spectral line.

Iron in the disc, illuminated by hard X-rays from above it, fluoresces at 6.4 keV in its own rest frame. The photons that reach an observer have been shifted twice: by the orbital motion of the gas that emitted them, which is a fraction of the speed of light, and by climbing out of the hole’s potential well. The first is Doppler and goes both ways; the second is a redshift only, and it grows without limit as the emitting radius approaches the horizon.

$$g = \frac{E_{\rm obs}}{E_{\rm em}} = \frac{\sqrt{1 - 3GM/rc^2}}{1 + \sin i\,\sin\phi\,\sqrt{GM/rc^2}}$$

with $i$ the disc’s inclination and $\phi$ the azimuth of the emitting patch. The numerator carries the gravitational redshift and the orbital time dilation together and is the same everywhere on a ring; the denominator carries the Doppler shift and changes sign across the ring.

The resulting line is not a line. It is a broad, skewed feature with a sharp blue edge from the approaching side of the outer disc and a long red tail from the innermost material — and the extent of that tail is set by how far in the disc goes.

The illuminating source itself is not resolved either; its size is known only from how long the disc takes to respond to it. The measurement is therefore of the faintest part of a broad feature, sitting on a continuum whose shape is itself being fitted. That is why spin values are argued about. The red wing has to be separated from absorption by intervening material, from a warm absorber in the source itself, and from the possibility that the disc simply does not extend to the last stable orbit. Different groups fitting the same spectrum with different continuum models have returned spins differing by more than the quoted errors.

The assumption both methods share #

The line method and the continuum method disagree about individual objects and agree about one thing, and what they agree about is an assumption rather than a measurement.

Both work by locating the inner edge of the disc and identifying it with the last stable orbit. The line method finds the edge from how far the red wing extends; the continuum method finds it from the temperature of the hottest annulus. Neither observes the last stable orbit; both observe where the emission stops, and then assume that is the same place.

The assumption is defensible. Inside the last stable orbit material plunges inward on a near-radial trajectory, spending very little time there and radiating comparatively little, so the emission does fall off sharply — and the sharpness is what makes the edge detectable at all.

It is also known to fail in one regime. At low accretion rates the inner disc is thought to evaporate into a hot, thick, radiatively inefficient flow, truncating the disc at tens or hundreds of gravitational radii — far outside the last stable orbit, and at a radius that has nothing to do with the spin. A spin fitted to such a spectrum returns a low value regardless of what the hole is doing, and the low-spin measurements in the literature are exactly the ones this worry attaches to.

The defence is to measure the same object at different accretion rates. If the fitted inner radius stays put while the luminosity changes by a factor of ten, it is a property of the metric; if it moves, it is a property of the flow. That test has been done for a handful of X-ray binaries in outburst, and for those the radius holds constant over a wide range — which is the strongest evidence available that the edge being measured is the one the theory names.

The measurement is therefore contested at the level of what is being seen rather than how well, and that is an unusual position. Improving the data does not settle it; the argument is about whether the model connecting the data to the spin applies to that source at that time.

What an image of one would say #

A third route arrived with the ability to resolve a horizon-scale structure directly, and it is worth stating what it does and does not deliver.

Light passing close to a black hole can orbit it. The set of trajectories that circle before escaping produces a thin bright ring, and inside it a region from which no light reaches the observer — the shadow. Its size is set almost entirely by the mass: for a non-rotating hole the shadow’s diameter is $2\sqrt{27}\,GM/c^2$, and for a maximally rotating one seen from the pole it is barely different.

That near-independence is the important part, and it runs both ways. It makes the image an excellent mass measurement and a poor spin measurement: the shadow’s diameter varies by only a few per cent across the whole range of spin, which is comparable to the calibration uncertainty of the instrument.

What does depend on the spin is the shadow’s shape — it becomes asymmetric, flattened on the side where the frame dragging runs against the light — and the brightness distribution around the ring, which is strongly asymmetric because material on the approaching side is beamed towards the observer. Both are second-order effects on an image that is at the resolution limit, and extracting a spin from them requires a model of the emitting plasma that is at least as uncertain as the continuum models the X-ray methods argue about.

So the image has not settled the spin question and was not going to. What it settled is a prior one: that the object has a horizon-scale structure of the size and shape the metric predicts, which is the assumption every spin measurement was already resting on.

There is a fourth handle now being tried, and it is the first genuinely new one in twenty years. X-rays reflected from a disc are polarised, and the degree and angle of that polarisation depend on the geometry of the reflecting surface and on how much the light’s plane of polarisation is rotated on its way out of the strong field. Measuring it constrains the inclination and the inner radius separately, which is exactly the degeneracy that makes the spectral fits ambiguous. The first results from an X-ray polarimeter in orbit have already forced revisions to the assumed geometry of the illuminating source in several sources, which is a promising sign in the same way an inconvenient result usually is.

It is worth being explicit about what a spin measurement is being used for, because the requirement is looser than the debates suggest. Almost nothing in astrophysics depends on knowing one hole’s spin to a few per cent; what the subject wants is the distribution over a population, and the difference between a distribution clustered near the limit and one spread across the range survives systematic errors that would ruin an individual measurement. That is a fortunate situation, and it is why the field has continued to publish contested numbers rather than waiting for a method nobody argues about.

The same is true of the mass, for a different reason: masses agree between methods to within a factor that would be embarrassing for a stellar measurement and is entirely adequate for deciding how a population grew.

The limit that is not one #

There is a natural expectation that a hole fed for long enough by a disc will reach the extremal spin, since every parcel of material arriving from a prograde disc brings angular momentum. It does not.

The physical limit is lower still, and it comes from the light. Photons emitted by the disc do not all escape; some are captured by the hole, and the capture cross-section is not symmetric — a photon on a retrograde path is more easily swallowed than a prograde one. The hole therefore absorbs a net negative angular momentum from its own disc’s radiation, and that spin-down balances the accretion spin-up at $a \approx 0.998$.

The number is model-dependent — it assumes a thin disc radiating efficiently — and a thick disc that advects its energy inward rather than radiating it does not impose the same limit. But it does mean that a hole measured at a spin very close to one is telling a story about how it was fed, and a hole measured at a low spin is telling a different one.

Both of the curves the measurement rests on are worth reading over a narrower range of spin, where nearly every real hole sits.

What a distribution of spins would say #

This is the reason the measurement is worth the difficulty. Spin is a fossil of assembly.

A hole that grew by swallowing a long-lived disc, always fed from the same direction, spins up: after doubling its mass it is turning fast, and after growing by a factor of two and a half it is at the limit. A hole that grew by a long series of mergers with randomly oriented partners does not: each merger adds angular momentum in a random direction, and a random walk in three dimensions accumulates as the square root of the number of steps while the mass accumulates linearly. Such a hole settles near $a \approx 0.7$ and then drifts down.

Testing that there are only two #

The claim that a hole has exactly two numbers is a theorem about a particular theory, and it is testable.

When two holes merge, the object left behind is initially not a black hole in equilibrium — it is a distorted horizon, and it settles by radiating away its distortions. The radiation is a superposition of damped sinusoids, and the frequencies and damping times of those modes are fixed, in general relativity, by the final mass and spin alone. Measure two of them and the mass and spin follow; measure a third and the theory is being tested rather than applied. The spins measured this way — from the ringdown, and from the inspiral’s dependence on the two bodies’ spins — are the second independent route to the number, and they apply to holes of a few tens of solar masses rather than to the millions the X-ray methods reach. The two populations mostly do not overlap, so the agreement between them is a check on the framework rather than on any object.

What the merger measurements have delivered so far is mostly a statement about the component spins before the merger, and it is a negative one. The waveform is sensitive to a particular mass-weighted combination of the two spins projected along the orbital axis, and that combination is consistently small — clustered near zero, with a modest tail. It is a poor constraint on each hole’s spin separately, because a large spin lying in the orbital plane contributes almost nothing to the measured combination, but it is a good constraint on the aligned component, and for the isolated-binary channel the aligned component is what the spins ought to be.

That is awkward in the same way the accretion argument above is instructive. A hole formed from a massive star’s collapsing core inherits that core’s angular momentum, and a core that had not been braked would produce a hole spinning near the limit. The observed near-zero spins therefore say that the transport which keeps a red giant’s core turning far more slowly than conservation allows is still operating in the progenitors of black holes — which is the same unexplained efficiency, measured on a completely different object, by a completely different instrument.

Two more line profiles show how much of a spin measurement is a statement about the disc rather than about the hole.

Where the ladder goes #

The nearest rung is the assembly problem the spin distribution is evidence about: what has to happen before two holes at the centre of a merged galaxy can reach each other at all, which is an angular momentum problem of a completely different kind and one that gravity alone does not solve.

The other direction is towards what spin does besides deepening the well. A rotating hole threaded by a magnetic field can have energy extracted from its rotation directly, without anything falling in — the field lines are dragged round, and the resulting electromagnetic torque carries away rotational energy at a rate proportional to the square of the spin and the square of the field. That is the leading candidate for the jets that emerge from active nuclei at nearly the speed of light, and it is the one process in astrophysics where the energy source is not mass and not fusion but the rotation of empty space.