The circle light cannot leave

The horizon is defined by what cannot come back from it. That makes it a statement about the future rather than about a surface anybody can see, and it is not the radius a photograph of a black hole shows.

There are three radii, they are different, and each is the answer to a different question.

The potential light moves in, and what it lacks #

A particle orbiting a mass moves in an effective potential: the real gravitational attraction plus the centrifugal term that comes from its angular momentum. In Newtonian gravity that combination has a minimum, and the minimum is what circular orbits are. In the Schwarzschild geometry it has a minimum and a maximum, and the innermost stable circular orbit is where the two merge.

For light there is no minimum at all.

The function rises from nothing far away, peaks once, and falls into the horizon. Its peak sits at $1.5$ horizon radii — three masses in the units where the horizon is at two — and it is found here by searching the drawn curve rather than differentiated, which is why the figure prints $1.5000$ rather than asserting it.

A maximum is an unstable equilibrium. So light can travel in a circle at three masses, and a ray doing so leaves at the smallest disturbance: inward, and it falls in; outward, and it escapes. Nothing accumulates there, and there is no shell of trapped light.

The number the peak’s height gives #

Whether a given ray clears the peak depends on its energy against the peak’s height, and for light the only thing distinguishing one ray from another is the ratio of its angular momentum to its energy — the impact parameter, which is how far off-axis the incoming ray would have passed if nothing had bent it.

Setting the ray’s energy equal to the peak’s height gives the critical value directly:

$$b_c = \frac{\sqrt{27}}{2}\,r_s = 3\sqrt3\,\frac{GM}{c^2} = 5.196\,\frac{GM}{c^2}.$$

Rays aimed closer than that go over the peak and are captured. Rays aimed further are turned back, however deeply they dip in on the way.

That number is the one a distant observer actually measures, because it is the radius of the region on the sky from which no light arrives. A black hole’s shadow is $2.6$ times the horizon radius, not one times it, and the difference is entirely the bending suffered by the light that just escapes.

The three radii, then: $2M$ for the horizon, $3M$ for the photon sphere, $5.196M$ for the shadow’s edge. Every one of them is sometimes called the size of a black hole and they differ by a factor of two and a half.

It is worth writing the three radii as one sentence, because the ratios are fixed for every non-rotating black hole there is: the horizon at $2M$, the photon sphere at $3M$, the shadow’s edge at $3\sqrt3\,M = 5.196M$. Nothing about the mass enters except as the scale, so a hole of a billion suns and one of ten have shadows in exactly the same proportion to their horizons, differing only in how large the whole thing is.

What the rays do near the critical value #

The traced geodesics make the transition visible, and it is not a gentle one.

At three per cent below the critical parameter a ray falls in. At two per cent above it, the ray traced here wraps $1.06$ times round the hole before leaving — it comes back out having gone past the far side and round again.

Take the impact parameter closer still to critical and the number of turns grows without limit, logarithmically: each extra turn costs a factor of about $e^{2\pi}$ in how finely the parameter must be tuned. What that produces on the sky is an infinite sequence of images.

Light from a source behind the hole can reach the observer having gone round zero times, once, twice, and so on, and each of those paths has its own impact parameter, converging on $b_c$ from outside. So the shadow’s rim is encircled by a photon ring: a bright, thin feature made of light that went round at least once, with each successive subring a factor of $e^{-2\pi}\approx0.002$ fainter and closer to the edge.

That factor of $e^{2\pi}$ is not a coincidence of the Schwarzschild solution. It is the Lyapunov exponent of the unstable circular orbit, expressed per orbit, and it is the same kind of number an unstable orbit anywhere gives.

The infinite sequence has a practical consequence that is easy to state and hard to observe. Because each subring is a fixed factor fainter and a fixed factor closer to the critical radius, their spacing on the sky is a pure number of general relativity — independent of the mass, the spin’s effect aside, and independent of every detail of the gas producing the light. Measuring that spacing would be a test of the geometry with no astrophysics in it at all, which is why it is the target of the next generation of such measurements, and why the first ring’s brightness profile is being studied so closely for the shoulder the second one should put on it.

The same equation, run further out #

The strong-field behaviour above and the 1.75 arcseconds measured at the Sun’s limb are not two theories with a boundary between them.

The equation integrated for every ray in this essay is

$$\frac{d^2u}{d\phi^2} + u = 3Mu^2, \qquad u = \frac1r,$$

and the whole difference from the Newtonian problem is the term on the right. Drop it and the solutions are conic sections and nothing is ever captured. Keep it and it is negligible wherever $r \gg M$, which includes every ray that has ever grazed a star.

Running the same integration at an impact parameter of sixty masses gives a deflection of $0.0666$ radians against the weak-field $4M/b = 0.0667$. The strong field is not a different regime; it is where a term that is usually a millionth stops being one.

What a photograph shows #

Everything so far concerns a non-rotating mass and light from far away. What an image of a real black hole shows is a mixture of that geometry and a great deal of astrophysics.

Nothing in this essay depends on the mass except through one scale. The horizon radius is proportional to the mass, and the photon sphere and the capture radius are fixed multiples of it — one and a half, and about 2.6 — for every black hole there is. So a supermassive hole and a stellar one produce geometrically identical pictures at different sizes, and a photograph of either shows the same ratios. That is why an image of one object constrains the theory for all of them.

The dark patch is the shadow, and its outline is the critical impact parameter — a purely geometrical prediction with no astrophysics in it. The bright ring around it is emission from hot gas, whose brightness distribution depends on the accretion flow, the magnetic field and the viewing angle, and which is where nearly all the modelling difficulty lies.

The two are worth separating because only the first is a clean test. The shadow’s size, given an independently measured mass and distance, is a prediction of general relativity to within a few per cent; the ring’s brightness and asymmetry are a measurement of the gas.

Rotation changes the shadow’s shape as well as its size. For a maximally rotating hole seen edge-on, the shadow is flattened on the side where the horizon is moving towards the observer, and the critical impact parameter differs between prograde and retrograde light. The difference is at the ten per cent level and is the next thing such measurements are aimed at.

Signals sent outward at intervals by something falling in arrive stretched out and reddened without limit, and that is what a shadow’s rim is made of. The light forming it left long ago, from close in, and has been climbing since — so the edge of the dark region is not a surface anyone is looking at but the accumulated late arrivals of everything that ever fell.

The area that swallows light #

The critical impact parameter is a length on the sky, and squaring it gives an area — the cross-section a black hole presents to light coming from far away.

$$\sigma = \pi b_c^2 = 27\pi\,\frac{G^2M^2}{c^4}$$

which is $6.75$ times the area of the horizon as measured by its own radius. A black hole is a considerably better absorber than its horizon is large, and by a factor that is a pure number: it does not depend on the mass, on the wavelength, or on anything else.

How much of a distant source’s light a given cross-section intercepts is fixed by the inverse square, and the capture cross-section here is the area to be intercepted. It is larger than the horizon — the geometry bends rays inward, so a hole swallows light aimed at a circle wider than itself — and the ratio between the two areas is a pure number that depends on nothing but the geometry.

The same question for a massive particle has a different and stranger answer. A slow-moving particle has a much larger capture cross-section than light does, because its trajectory is bent more for the same closeness of approach, and as the speed goes to zero the cross-section diverges — anything drifting slowly enough is captured from any distance, given time. As the speed approaches that of light the cross-section falls to the photon value, which is its floor.

That divergence at low speed is why accretion is easy and radiation pressure is a real competitor to it. A luminous body can push matter away faster than gravity pulls it in, and the balance between those two is what sets the maximum rate anything can accrete.

What the sky looks like from close in #

The three radii can be turned round and read as statements about what an observer at each of them would see.

Hovering close is expensive in exactly the currency of clock rates. The factor by which a hovering observer’s clock runs slow is the same one that appears in every other quantity here, and it diverges at the horizon — so staying put close in requires an acceleration that grows without bound, and the sky seen from there is compressed into a bright spot overhead.

Far away the black hole is a small dark disc of angular radius set by $b_c$ over the distance. Move inward and the disc grows, faster than the geometry of a solid object of that size would grow, because the light from behind the observer is being bent round.

At the photon sphere the black hole covers exactly half the sky. That is a clean statement and a useful one: an observer hovering at three masses sees a hemisphere of darkness and a hemisphere containing the entire rest of the universe, compressed into it. Below the photon sphere the darkness covers more than half, and the whole external universe is squeezed into a shrinking cone overhead — which closes to a point as the horizon is approached.

The compression is the same effect as the aberration that crowds a fast traveller’s sky into a cone, arrived at by a different route: there it comes from the observer’s motion, here from the geometry, and in both cases the sky’s brightness is concentrated with it.

Hovering there is not free. Staying at rest at three masses requires an acceleration that a rocket has to supply, and the clock of anyone doing so runs slow by a factor of $\sqrt{1-2/3}$, which is $0.577$. At the horizon that factor goes to zero and hovering becomes impossible at any acceleration.

The same circle, heard rather than seen #

The unstable orbit at three masses has a second observable consequence, and it arrives by a completely different instrument.

Disturb a black hole — by dropping something in, or by merging it with another — and it rings. The disturbance settles by radiating gravitational waves at a set of complex frequencies fixed by the hole’s mass and spin and by nothing else: a tone that decays, with a pitch and a decay time that are properties of the geometry.

Those frequencies turn out to be the photon sphere’s. In the limit of short wavelengths a gravitational wave is itself following null geodesics, and a wave that lingers is one that has been trapped near the maximum of the same effective potential this essay is about. The real part of the ringing frequency is the orbital angular frequency of the circular null geodesic, and the imaginary part — the rate at which the tone dies away — is the Lyapunov exponent of that orbit, the very number that makes each photon subring fainter than the last by $e^{-2\pi}$.

That the two coincide is worth stating as arithmetic rather than as a slogan. For a non-rotating hole the Lyapunov exponent and the orbital frequency are equal, so the ringdown loses a factor of $e^{-2\pi}$ per cycle for the lowest modes — the same factor, from the same instability, as the ratio between successive subrings on an image.

The numbers are within reach of existing instruments. A sixty-solar-mass remnant rings at a couple of hundred hertz and falls silent in a few milliseconds, which is exactly what a gravitational-wave detector recorded at the end of the first binary black hole merger observed. So the photon sphere has been measured twice: once as the rim of a dark patch at radio wavelengths, and once as the pitch of a tone lasting a thousandth of a second, with nothing in common between the two experiments except the circle.

The view from the circle itself #

The statement that a black hole covers half the sky at three masses has a consequence that is worth following to its end, because it is the sharpest available illustration of what an unstable light orbit means.

An observer hovering exactly at the photon sphere, looking exactly along the tangent, is looking down the track of a circular null geodesic. Light leaving them in that direction goes right round and comes back — so they see, in the sideways direction, the back of their own head, at a delay of about a millisecond for a ten-solar-mass hole. And not one image but an infinite sequence, from light that went round twice, three times, and so on, each fainter by the usual factor and all lying in the same direction.

Since the tangential direction is a whole circle of directions, the image is not a point but a ring around the observer’s own sightline — a self-portrait in the shape of an Einstein ring, at the boundary between the dark half of the sky and the bright one.

It is also the least robust observation in physics. The orbit is unstable, so a displacement of any size at all sends the light either inward to the horizon or outward to infinity, and the ring is destroyed rather than merely blurred. What makes the photon sphere visible from far away is precisely that it does not have to be occupied: the shadow’s rim is made of rays that came close to the peak and left, and those are plentiful, while the rays that stay are a set of measure zero.

Where the model runs out #

The mass is not rotating and every astrophysical black hole is. Angular momentum splits the photon sphere into a range of radii depending on the direction of travel, drags the light around with the hole, and turns the shadow from a circle into a shape with a flattened side. Everything quantitative here is the zero-spin limit.

Light is treated as a ray with no wavelength. The geodesic description holds while the wavelength is small compared with the mass’s own length scale, which for a stellar-mass hole means anything shorter than kilometres. It fails for radiation of wavelength comparable with the hole, where the capture cross-section acquires structure and the sharp critical parameter is replaced by a gradual transition.

Nothing here is emitting. The shadow is defined against a background of light coming from somewhere else, and its visibility depends entirely on there being such a background. A black hole in an empty universe casts no shadow because there is nothing for it to be dark against.

And the observer is at infinity. The critical impact parameter as stated is the asymptotic one. An observer at a finite distance sees a shadow of a different angular size, and one close in sees a very different sky indeed — at the photon sphere itself, the black hole covers exactly half the sky, and below it more than half.

Everything in this essay is classical geometry, and the one thing it cannot describe is why a horizon is not perfectly black after all. A horizon has a temperature, that temperature falls with mass, and its existence has nothing to do with the ray-tracing here — it comes from quantum field theory in curved spacetime, and it is the boundary of what these figures can say.

How the numbers were known before anything could be seen #

The photon sphere and the critical impact parameter were computed within a few years of the metric being written down, and stayed a curiosity for most of a century.

Hilbert obtained the exact deflection integral in 1917, and the value $3\sqrt3\,GM/c^2$ follows from it immediately. What took much longer was the recognition that it describes something observable, because for a stellar-mass black hole at any plausible distance it subtends an angle far below anything that could be resolved.

The change came from the mass rather than from the optics. A black hole of a few billion solar masses at the centre of a nearby galaxy has a shadow of some tens of microarcseconds — still minute, but reachable by an interferometer with a baseline the size of the Earth. Interferometry measures an angle from a correlation rather than from an image, so the resolving power is set by the separation of the telescopes rather than by the size of any of them.

The reconstruction that results is not a photograph, and the distinction matters for what has been tested. What is measured is a set of correlations between widely separated antennas, from which a brightness distribution consistent with them is reconstructed. The dark central region and its size are robust across reconstruction methods; the details of the ring’s structure are less so.

What has been confirmed is the geometrical part: the dark region’s angular size, given a mass measured from stellar orbits and a distance measured independently, is what $b_c$ predicts. That is the same prediction as the deflection at the Sun’s limb, at a field strength a hundred million times greater, and it is the only place the strong field has been tested optically.

The ladder from here #

Later rungs on this anchor: the photon ring’s subrings and what measuring their spacing would test; the shadow of a rotating hole and the asymmetry that measures the spin; the capture cross-section for massive particles, which is larger than for light and depends on the speed; and the interior, where the radial coordinate becomes timelike and the singularity is a moment rather than a place.

The neighbouring ladders are the surface that only lets things in, which is the horizon itself, the horizon that nothing marks, which is about what a falling observer notices there, and the bend Newton got half right, which is the same deflection where the extra term is small.