A ring whose size is almost only a mass

Both of the spectral routes to a black hole’s spin measure one length — where the accretion disc stops — and convert it through one relation. Everything that can go wrong with them goes wrong at the same place: they assume the disc reaches the last stable orbit, and nothing in a spectrum tests that assumption.

An image is a different instrument entirely. It does not need a disc to end anywhere in particular, because the boundary it draws is made of light rather than of matter, and light has its own innermost orbit several gravitational radii inside the one matter can hold.

The orbits nothing is on #

A photon aimed close enough to a black hole is captured and a photon aimed far enough away escapes, and between the two there is a family of orbits that do neither: they circle the hole indefinitely at a fixed radius, unstably, so that the smallest perturbation sends the photon in or out.

Those orbits are what makes a shadow. Every direction on a distant observer’s sky corresponds to a ray traced backwards, and each ray either reaches out to the sky beyond or ends on the horizon. The set of directions whose rays end on the horizon is a patch, and its boundary is the set of directions whose rays asymptote to one of the unstable orbits. That boundary is the critical curve, and it is a statement about the geometry rather than about anything in it.

For a hole that does not rotate the arithmetic is one line. The unstable circular photon orbit sits at three gravitational radii, and a photon on it has an impact parameter, as measured by a distant observer, of $b = 3\sqrt{3}\,GM/c^2 = 5.196\,GM/c^2$. The shadow is therefore a circle of exactly that radius — $\sqrt{27}$, a number with no free parameter in it — and it is larger than the horizon, which is at two. A hole’s dark patch is more than two and a half times the size of the hole.

That is the first thing the picture gets wrong if it is read naively. The shadow is a lensing effect, not a silhouette. The rays that define its edge have wound round the hole and come back out, so the boundary is the image of a surface that the horizon has been magnified into, and the magnification is a factor of 2.6.

Why the size is almost entirely a mass #

The interesting number is what happens to $\sqrt{27}$ when the hole is spun up.

Seven per cent, against a factor of six. The two quantities are computed from the same metric and they behave in opposite ways, and it is worth being clear about why, because the contrast is the whole content of this essay.

The last stable orbit is where a massive particle’s effective potential loses its minimum, and frame dragging moves that point a long way: a co-rotating particle can use the circulation of spacetime to stay in a stable orbit much closer in. Photons do not have the same freedom. The prograde photon orbit does move inward with spin, from three radii to one, but the impact parameter that corresponds to it — which is what an observer measures, since the shadow is an angle on the sky and not a distance near the hole — barely changes, because the orbit that shrinks is also the orbit whose light climbs out of a deeper well and is bent more on the way.

The two effects nearly cancel, and the near-cancellation is not a coincidence in the sense of being a numerical accident. It is a statement that the capture cross-section for light is set mostly by the mass, which is the quantity that sets the scale of the whole metric, and only marginally by the way that mass is rotating.

So a measured angular diameter, divided by a distance, is a mass. Nothing else has to be supplied — not an inclination, not an accretion rate, not a radiative model, not a virial factor of unknown value, and not a hardening factor from a disc atmosphere. The quantity of interest is a length, and the only thing standing between the measurement and it is a distance, which for a nearby galaxy comes from a stellar population and carries its own few per cent.

That makes it the cleanest mass in the subject. Every other route to a supermassive hole’s mass — the velocity dispersion of the stars around it, the width of a broad emission line, the orbit of an unresolved gas disc — reaches the mass through the motion of material whose geometry is assumed rather than seen.

Where the spin actually is #

The same insensitivity that makes the mass easy makes the spin hard, and the two are not separate facts.

The departure from circularity is the quantity that is quoted when a spin is claimed from an image, and the figure above is the honest statement of its size. It is small in absolute terms — a few per cent at best — and it collapses toward face-on, which is the orientation that the brightest, most nearly jetted sources tend to have and the orientation both published images were made at.

There is a larger spin signal, and it has a worse problem.

The displacement is the biggest thing spin does to the outline. Photons circling with the hole are dragged round and escape from much closer in; photons circling against it cannot come as close as they could around a hole with no rotation at all. So the capture region is lopsided, and the whole shadow sits off to one side.

A displacement measured against nothing #

The reason the largest signal is unusable is a sentence long and it is worth stating plainly: a displacement requires an origin, and nothing in the image marks one.

The shadow’s outline is displaced with respect to the hole’s position — the location of the mass itself, projected onto the sky. An interferometric image does not contain that position. Closure phases, which are what such an image is mostly made of, are blind to absolute position by construction: translating the whole source on the sky leaves every closure phase unchanged. That immunity is what allows an image to be made across ten thousand kilometres of atmosphere at all, and it costs exactly the quantity this measurement needs.

Even with a position, the reference would not help. What the array measures is the brightness of the accretion flow around the shadow, and locating the hole within that means deciding where the flow is centred — which depends on where the flow happens to be bright, which depends on the magnetic field, the electron temperature and the viewing geometry, all of which are model.

So the ordering is inverted in a way worth holding onto. The measurement that needs no reference at all — a size — carries almost no spin. The measurement that carries most of the spin — a displacement — needs a reference the technique is structurally incapable of providing. The spin ends up being extracted from the residual, the departure from circularity, which is the smallest of the three.

The asymmetry that was detected is a different one #

Both published images are conspicuously brighter on one side than the other, by a factor of two or three, and it is easy to read that as the lopsidedness the figures above are about. It is not. The geometric displacement is a property of the dark patch’s outline; the brightness asymmetry is a property of the flow, and the two carry different information.

Gas orbiting a hole at a substantial fraction of the speed of light is beamed: the same relativistic Doppler effect that shifts a line also concentrates the emission forwards, so the side of the ring where the gas is approaching is brighter and bluer and the receding side is dimmer. At the last stable orbit the orbital speed is of order half the speed of light, and the resulting ratio between the two sides is a factor of a few — far larger than any purely geometric effect on the outline.

That asymmetry is measurable and it was measured. What it says is which way the gas goes round, and, combined with the direction of the large-scale jet, which way the hole’s angular momentum points relative to it. It does not say how fast the hole turns, because the beaming depends on the orbital speed of the flow, which depends on where the flow is rather than on the metric alone — a ring of gas at eight gravitational radii around a slowly rotating hole and one at four around a fast one produce similar contrasts.

So the picture contains two asymmetries with two causes. One is enormous, easily detected and about the gas. The other is a few per cent, at the edge of what the array resolves, and about the hole. The conspicuous feature is the one that carries the less fundamental information, which is a recurring hazard whenever a geometric signal is observed through a radiating medium, and the same hazard the disc’s own inner edge presents to the spectral methods.

What a published image is, and what it is not #

The two images that exist were made by an array of eight to eleven sites observing at 1.3 millimetres, and everything the closure identity’s own account of image reconstruction says applies here with unusual force.

The transform plane is sampled in a handful of arcs rather than filled, so an image is chosen from a family consistent with the data rather than determined by them. The absolute phase is unrecoverable, so the reconstructions rest on closure quantities. And the array’s resolving power is comparable with the feature being measured: the synthesised beam is about twenty microarcseconds and the ring is about forty, so the picture is resolved by a factor of two rather than by a factor of ten.

The thing the analysis does well, and the reason the mass is trustworthy despite all of that, is that the quantity extracted is not read off a picture. It is fitted directly to the visibilities: the diameter of a ring is a spatial frequency at which the visibility amplitude has a minimum, and a minimum in a curve is a much more robust thing to measure than a feature in a reconstruction. The published angular diameter is a fit to the data; the image is an illustration of it.

That distinction matters for what can be claimed. A diameter fitted to the visibilities is a measurement. A departure from circularity of half a per cent, extracted from an image that a regulariser helped to make, is not.

What was actually measured #

For the nearby giant elliptical, the chain is: an angular diameter of 42 microarcseconds for the bright ring, a correction from the ring’s diameter to the critical curve’s that comes from simulations of the emitting plasma, a distance of 16.8 megaparsecs from the surface-brightness fluctuations of the galaxy’s stars — which is itself the last step of a chain every part of which was calibrated by the one before it — and the conversion.

Two of those four are not measurements of this object. The distance is an independent problem with its own few per cent, and the correction from the observed bright ring to the underlying critical curve is where the modelling lives — the emission is not on the critical curve, it is in a flow around it, and how much brighter the ring is on the outside than the inside depends on the flow.

The published number is a mass of $6.5 \times 10^9$ solar masses with a systematic uncertainty of about ten per cent, and the striking thing about it is the comparison rather than the value. The same galaxy’s centre had been weighed twice before, by the motions of its stars and by the motion of its gas, and the two answers differed by a factor of two. The image agrees with the stellar-dynamical one. That agreement is the actual result: a method with an entirely different systematic, applied to the same object, landing on one of two previously irreconcilable answers.

For the hole at the centre of this galaxy the chain is shorter in one respect and worse in another. The distance and the mass were already known to about one per cent, from decades of tracking individual stars on orbits around it — so the image was not needed for the mass and instead became a test of it, which it passed. What it cannot do is the spin, and for a specific reason: the orbital period at the last stable orbit is minutes rather than days, so the source varies faster than the observation takes, and the standard assumption that the sky is static during a synthesis is simply false.

Where the model stops #

The shadow is not the horizon. It is 2.6 times larger, and its size is the capture cross-section for light rather than any surface. Nothing in an image resolves a horizon, and nothing in an image tests whether there is one — a compact object with a surface but no horizon would produce nearly the same dark patch, and separating the two rests on the absence of light from a surface rather than on the geometry of the ring — an argument from a non-detection, with all the difficulty that any error budget built out of upper limits carries.

The emission is not on the critical curve. The curve is infinitely thin and the ring is not. What is observed is a flow whose brightness peaks a little outside the curve, and the offset between the two is computed from simulations of a plasma whose electron temperature is a free parameter.

The inclination is not measured by the image. For the giant elliptical it is taken from the large-scale jet, which is assumed to lie along the spin axis. That assumption is the reason a spin can be discussed at all, and nothing in the picture confirms it.

And the third dark feature is not the point. Inside the critical curve there are further, exponentially fainter rings — light that wound round the hole once, twice, and so on before escaping — whose diameters approach the critical curve’s geometrically. The $n = 1$ ring carries a few per cent of the flux and is sharper than anything the array resolves. It is the object several proposed space missions are designed for, because its diameter is a still cleaner statement about the metric, and it is not in either published image.

The generalisation #

The shape worth carrying out of this is about which of two quantities to build a measurement on, and it is the opposite of the reading the spectral argument arrived at.

There, a stiff conversion turned out to be a good one: the last stable orbit’s steep dependence on spin near the extremal limit is what makes an extreme spin better determined than a moderate one. Here the same steepness is the problem in reverse. The shadow’s size is insensitive to spin, and that insensitivity is exactly what makes it a good mass — a quantity that does not depend on the unknown is a quantity that can be measured without it.

Both statements are the same statement about derivatives, read for different purposes. A measurement is limited by the derivative of the observable with respect to what is not being measured, and helped by its derivative with respect to what is. The shadow has a large one with respect to mass and a small one with respect to spin, which is why the mass comes out clean and the spin does not — and why an instrument designed to measure the second would have to be built around the faint interior rings rather than around the outline.

There is a second reading, about references. Three quantities were available in the picture and they differ in what they need to be measured against: a size needs nothing, a shape needs nothing, and a position needs an origin. The technique that made the image is one that deliberately discards positions. Whenever a method’s immunity to a systematic is structural, the quantity it is immune to is a quantity it has given up, and the thing to ask of any clever differential measurement is what it can no longer see.

Still open: whether a ring can be read for spin at all #

The faint interior rings are where this goes next. Their diameters converge on the critical curve’s geometrically, their shapes are set by the same two numbers, and their relative diameters depend on spin much more strongly than the outline does — which makes them the one feature in the picture whose spin sensitivity is not a residual. Resolving them requires baselines longer than the Earth.

Beside them sit the two other routes nothing here has taken: the quasi-periodic oscillations, whose frequencies would be a spin measurement if anybody knew which resonance produced them, and jet power against spin, which is the Blandford–Znajek prediction and the least conclusive of the tests. And beside those, the measurement that observes spin as an angular momentum rather than through anything it does to light — which is the next thing to take up, and the only route with no accretion flow in it anywhere.