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Table I. Where you are
Distance from the centre
Your speed inward
Your watch, seen from home
Home’s clock, seen by you
Stretch, head to foot
Left on your watch
Your green laser, at home

Table I. Where you are

Distance from the centre

Your speed inward

Your watch, seen from home

Home’s clock, seen by you

Stretch, head to foot

Left on your watch

Your green laser, at home

A descent in nine stopsr = 100 r_s

Into a Black Hole

You are falling, feet first and freely, toward . Nothing in this picture is painted. Each frame is worked out by following light backward from your eye, along the curved paths that gravity gives it.

The black disc in the middle is not a thing you are seeing. It is the patch of sky that no light comes from. The glowing band is a flat disc of hot gas in orbit round it, and the stars are behind.

Distances are counted in Schwarzschild radii, written r_s: the size of the hole itself. For this one, r_s is , and you are out. Pick another hole in Table I at any time: the picture stays the same and the numbers change.

Scroll to fall

§ 1r = 58 r_s

Far away, it is only gravity

A black hole does not suck. Out here it pulls exactly as hard as a star of the same mass would. Swap the Sun for a black hole of one solar mass and the Earth would carry on in the same orbit, in the dark.

What gives it away is the sky behind. Light from distant stars passes the hole on its way to you and is bent toward it, so each star appears pushed outward from where it really is. Stars almost exactly behind the hole are smeared into arcs. A star directly behind would become a full circle, called an Einstein ring.

bend ≈ 2 r_s / b(1)

Equation (1) is the far-away rule: light passing at a distance b is turned through that angle, in radians. Twice as close, twice the bend. Near the hole the rule fails and the paths have to be computed, which is what this page does.

§ 2r = 30 r_s

How to read the picture

The disc is flat and thin, like Saturn’s rings, and you are looking at it almost edge-on. A flat ring seen edge-on should be a line. Yet it stands up in an arch over the hole, and a second loop hangs underneath.

Both are the far side of the disc. Light leaving its top surface heads upward, is bent over the hole and comes down to you: the arch. Light leaving its underside is bent the other way round: the lower loop. You are seeing the back of the disc from above and from below at once.

The left side is brighter and bluer than the right. The gas there is coming toward you at a large fraction of the speed of light, which squeezes its light to shorter waves and beams it forward. On the right it is going away, and is dimmer and redder.

§ 3r = 13 r_s

The disc

Gas cannot fall straight in. It arrives with sideways motion and goes into orbit, and the inner orbits run faster than the outer ones. The rubbing between them heats the gas until it glows, and lets it sink slowly inward.

By the time a kilogram of gas reaches the inner edge it has given up of its entire mass as light. Fusion in the Sun manages 0.7%. That is why a disc round a giant black hole, a quasar, can outshine all the stars of the galaxy it sits in.

About the colours. They are a temperature code, as for glowing metal: red is coolest, then orange, white, blue. The pattern is right but the scale is moved down so that it lands in visible light. A real disc round would be .

§ 4r = 3 r_s

The last stable orbit

You are passing the inner edge of the disc. Round a star, a planet can orbit at any height it likes. Round a black hole there is a lowest circular orbit that will hold: three Schwarzschild radii out. Gas there is moving at of the speed of light.

Inside it, no steady orbit exists. The slightest nudge inward and the gas drops, in a quick spiral, without having time to glow. That is why the disc has a sharp inner rim and a dark gap inside it.

§ 5r = 1.5 r_s

Where light itself can orbit

At one and a half radii gravity bends light so hard that a beam aimed exactly sideways goes round in a circle. This is the photon sphere. The orbit is a knife edge: a hair outside and the light spirals away, a hair inside and it spirals in.

It also explains the size of the dark patch. Seen from far away, any light ray aimed within 2.6 radii of the centre ends up inside this sphere and is lost. So the shadow looks 2.6 times wider than the hole really is, and its rim is a thin bright ring made of light that went round once, twice or more before escaping.

The dark patch ahead of you now is across. It has not swallowed the sky, because you are falling: your own speed, of light’s, crowds the view forward.

§ 6r = 1.12 r_s

Two clocks

Suppose you flash a lamp toward home once a second by your own watch. Each flash has to climb out against the inflow, and each starts from deeper than the one before, so it takes longer. Home sees the gaps between flashes grow: one second, then two, then a minute, then a year.

The same stretching happens to the light waves themselves. Your green laser arrives red, then as infrared, then as radio, and fades. Table I shows the colour it has at home right now.

The flash you send as you cross the horizon never arrives. Home never sees you go in. It sees a dimmer, redder, slower you, frozen just outside.

§ 7r = r_s

The horizon

You have just crossed it. Nothing happened.

There is no surface here, no wall, no flash. For the stretch across your body at this moment is . The horizon is not a place you can feel. It is the place where the last way out closes.

One way to picture it: space itself is flowing into the hole like a river toward a waterfall, faster and faster, and light is a fish that swims at one fixed speed. Here the river reaches that speed. Light aimed straight outward now hangs still. One step further in, it is carried down with you.

The view has turned to look sideways. Behind you the universe is still there, and you still see it, because its light falls in after you. Its clocks do not race: looking straight back, you see home’s clock running at of normal speed.

§ 8r = 0.5 r_s

Inside

Out in ordinary space you can stand still but you cannot stop time. In here it is the other way about. No rocket can hold you at this distance, however powerful, because the centre is no longer a place ahead of you. It is a time in your future, as unavoidable as next Tuesday.

For , you have left by your watch.

The outside sky is being squeezed into a band around your waist. Toward the hole it is crushed into a brilliant blue rim: that starlight is falling in too, and you are meeting it head on. Straight behind you it is dim and red.

§ 9r → 0

The centre

Gravity is stronger at your feet than at your head, so you are being stretched, and squeezed from the sides. Near the centre this difference grows without limit. For it passes ten times Earth’s gravity .

The dark now fills half of everything, and the last of the sky is a thin bright line.

Einstein’s theory says that all the mass that ever fell in is crushed here to a point of no size at all. Physicists read that answer as the theory failing, the way a calculator shows an error when asked to divide by zero. What replaces it needs a theory of gravity for the very small, and nobody has one yet.

Notes on the fall

What changes with the size of the hole, how anyone knows that black holes are real, what is still not known, and what this page left out.

Table II. Three black holes

The picture you scrolled through is the same for every non-spinning black hole, because every length in it scales with the mass. The experience does not scale. A small hole is far more violent at its horizon than a giant one.

How we know they are real

1916

Karl Schwarzschild, serving on the Russian front, solves Einstein’s new equations for the space round a single mass. The solution contains a radius, now named after him, inside which nothing can get out. Hardly anyone thinks nature would allow it.

1939

Robert Oppenheimer and Hartland Snyder show that a heavy enough star, once its fuel runs out, must collapse through that radius.

1971

Cygnus X-1, a bright source of X-rays, turns out to be a blue giant star circling something it cannot see. The unseen thing is now weighed at about 21 Suns: far too heavy to be anything but a black hole. The X-rays come from its disc.

1990s on

Two teams follow single stars whipping round the centre of our galaxy. One, called S2, takes 16 years per lap and reaches 7,700 km a second. Whatever it orbits has the mass of 4.3 million Suns packed into a space smaller than the Solar System. This won the 2020 Nobel Prize.

2015

The LIGO detectors feel space itself shake as two black holes, about 36 and 29 Suns, spiral together and merge, 1.3 billion light-years away. Three Suns’ worth of mass leaves as gravitational waves in a fifth of a second.

2019

The Event Horizon Telescope, radio dishes across the world working as one, photographs the shadow of M87*: a dark patch in a lopsided ring, the size and shape Einstein’s theory gives. In 2022 it does the same for Sagittarius A*.

What is not known

What is at the centre. The “point of no size” is where the theory stops working, not a description of anything.

Whether black holes last for ever. In 1974 Stephen Hawking showed that, once quantum physics is included, a black hole should glow very faintly and slowly lose mass. The glow of would have a temperature of above absolute zero, much colder than empty space, so today it gains more than it loses. Left alone it would take to vanish. Nobody has detected this glow.

What happens to what falls in. If a hole does evaporate, the details of everything it swallowed seem to be erased, and quantum physics says that cannot happen. This is the information paradox, and it is unsolved.

The far inside of a real hole. Real black holes spin, and the inside of a spinning one is expected to differ from the simple fall shown here.

What is simplified

The hole does not spin. Real ones do, some very fast. Spin drags space round with it, lets the disc reach closer in, and flattens one side of the shadow.

The disc is a flat, thin, opaque sheet ending exactly at the last stable orbit. Real discs have thickness, a hot haze above them and often jets. The two holes that have been photographed are fed by thin, puffy, glowing gas, not a neat disc like this one.

Disc colours are moved into visible light, as §3 says. The shifts between the blue and red sides are computed, not adjusted.

Brightness is compressed. The real approaching side would be hundreds of times brighter than the receding side. Here the difference is reduced so both can be seen, and a glow is added round bright things, as an eye or a camera would add.

The sky is invented. The stars, dust and coloured clouds are generated, and are shown far brighter beside the disc than they would look. Stars are drawn as points, brightened where the lens magnifies them.

Light is treated as arriving instantly. The picture ignores the minutes or hours that different rays take, so the disc’s swirl is drawn where it is, not where it was.

You are a point who started at rest very far away and fell straight in. A different path would give a different view at the same radius; someone hovering just outside the horizon would see the dark cover nearly the whole sky.

The fall inside the horizon follows Einstein’s theory with nothing added. Close to the centre that theory is not trusted.

How the picture is made

For the radius you are at, the page solves the equation for a light ray’s path, u″ = −u + 1.5 u² with u = r_s / r, for about a thousand directions of view, and keeps the results in a table. For every pixel a small program on the graphics card looks up where that ray crosses the disc, up to three times, and where on the sky it ends. Colour comes from the temperature of the gas there multiplied by the shift in frequency between the gas and you, which follows from the gas’s orbital speed, the depth of both in the hole’s gravity, and your own speed of fall.

Sources

K. Schwarzschild, “Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie”, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften (1916) 189–196.

J.-P. Luminet, “Image of a spherical black hole with thin accretion disk”, Astronomy and Astrophysics 75 (1979) 228–235. The first computed picture of a black hole with a disc.

O. James, E. von Tunzelmann, P. Franklin, K. S. Thorne, “Gravitational lensing by spinning black holes in astrophysics, and in the movie Interstellar”, Classical and Quantum Gravity 32 (2015) 065001. doi:10.1088/0264-9381/32/6/065001

A. J. S. Hamilton, J. P. Lisle, “The river model of black holes”, American Journal of Physics 76 (2008) 519. doi:10.1119/1.2830526

E. F. Taylor, J. A. Wheeler, E. Bertschinger, Exploring Black Holes, 2nd edition. The freely falling “rain” observer and the view from inside.

D. N. Page, K. S. Thorne, “Disk-accretion onto a black hole. Time-averaged structure of accretion disk”, Astrophysical Journal 191 (1974) 499. doi:10.1086/152990

Event Horizon Telescope Collaboration, “First M87 Event Horizon Telescope results. I”, Astrophysical Journal Letters 875 (2019) L1. doi:10.3847/2041-8213/ab0ec7

Event Horizon Telescope Collaboration, “First Sagittarius A* Event Horizon Telescope results. I”, Astrophysical Journal Letters 930 (2022) L12. doi:10.3847/2041-8213/ac6674

GRAVITY Collaboration, “Mass distribution in the Galactic Center based on interferometric astrometry of multiple stellar orbits”, Astronomy and Astrophysics 657 (2022) L12. doi:10.1051/0004-6361/202142465

B. P. Abbott et al., “Observation of gravitational waves from a binary black hole merger”, Physical Review Letters 116 (2016) 061102. doi:10.1103/PhysRevLett.116.061102

J. C. A. Miller-Jones et al., “Cygnus X-1 contains a 21-solar mass black hole”, Science 371 (2021) 1046. doi:10.1126/science.abb3363

S. W. Hawking, “Black hole explosions?”, Nature 248 (1974) 30. doi:10.1038/248030a0

C. Wyman, P.-P. Sloan, P. Shirley, “Simple analytic approximations to the CIE XYZ color matching functions”, Journal of Computer Graphics Techniques 2 (2013) 1–11. Used to turn temperature into colour.

Type: Old Standard, by Alexey Kryukov, under the SIL Open Font License: a revival of the faces that physics journals were set in when Schwarzschild wrote. Not a substitute for a textbook. Sizes and times are rounded to two figures.