- Black shadowRays that fell below the horizon. The shadow is about 2.6 times larger than the horizon itself.
- Photon ringA thin glowing line at the edge of the shadow: light that circled the hole once or more.
- Arc above the shadowThis is the far side of the disk! Its light bends over the hole. The underside of the disk shows up as a thin arc below the shadow.
- Brighter sideThere the gas rushes toward you at up to half the speed of light — the Doppler effect.
Time slows down
Far from the hole0 s
At the probe0 s
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The clocks run 10 times faster than real time.
Spaghettification
An astronaut (2 m tall) hangs at the same distance as the probe, measured in rs. Gravity pulls his feet harder than his head — that difference is the tidal force.
Calculator: horizon and evaporation
Enter a mass — we'll compute the horizon radius rs = 2GM/c², the temperature of the Hawking radiation and how long the hole would take to evaporate.
Horizon radius
Hawking temperature
Evaporation time
The star S2 around Sgr A*
Year
Distance from the hole
Speed
Animation: 1 year ≈ 1 s
The orbit in its true shape, seen “from above” (on the sky we view it at an angle). Period about 16 years, eccentricity 0.88, pericenter about 120 au — four times farther than Neptune is from the Sun. From the motion of S2 and other stars, the mass of Sgr A* was worked out: about 4.3 million solar masses in a region smaller than the orbit of S2. For this discovery Reinhard Genzel and Andrea Ghez received the 2020 Nobel Prize in Physics — together with Roger Penrose, who showed that black holes follow from Einstein's theory.
Three families
Mass scale: stellar black holes from a few to over a hundred solar masses, intermediate ones from a hundred to a hundred thousand, supermassive ones from a hundred thousand to billions of solar masses.
StellarFrom about 3 to a few dozen (rarely over 100) solar masses. They form when the core of a very massive star collapses. The first strong candidate: Cygnus X-1 — an X-ray source in the constellation Cygnus, identified in the early 1970s.
IntermediateFrom about 100 to 100,000 solar masses. The rarest and hardest to find; in 2020 the gravitational waves GW190521 revealed the birth of a hole of about 140 solar masses.
SupermassiveFrom hundreds of thousands to billions of solar masses, at the centers of galaxies: Sgr A* ≈ 4.3 million, M87* ≈ 6.5 billion solar masses. How they grew so fast is still unknown.
How a black hole is born
- A star of more than about 20 solar masses burns its fuel and ends its life in a supernova explosion (or collapses almost without one).
- If the leftover core is below the Chandrasekhar limit (about 1.4 solar masses) — we get a white dwarf (these form from lighter stars).
- A heavier core crushes electrons into protons, making neutrons: a neutron star forms. It holds up only to a limit (called the TOV limit) — by current knowledge about 2–3 solar masses; we still don't know the exact value.
- Above that limit nothing can stop the collapse: a black hole is born.
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More on the life and death of stars: Stars — from birth to supernova.
- September 14, 2015Both LIGO detectors (Hanford and Livingston) recorded the signal 7 ms apart. It was announced on February 11, 2016.
- 36 + 29 → 62 solar massesThe missing ~3 solar masses turned into wave energy: E = mc² ≈ 5·10⁴⁷ J, in a fraction of a second.
- Less than a protonThe wave changed the length of the detector's 4-kilometer arms by about 4·10⁻¹⁸ m — several hundred times less than the diameter of a proton.
- Nobel 2017Rainer Weiss, Barry Barish and Kip Thorne. Thorne also advised on the film “Interstellar” — the image of the black hole Gargantua was computed from his equations.
How does it work?
The physics in a few formulas — nothing to fear.
The event horizon
A black hole is a region from which nothing — not even light — can escape. Its boundary is the event horizon, with the Schwarzschild radius:
rs = 2GMc²G is the gravitational constant, M — the mass, c — the speed of light. The Sun would have to shrink to a ball about 3 km in radius, and the Earth — to about 9 mm.
How we compute the path of light
In the ray's plane of flight we describe it by the angle φ and the inverse of the distance, u = 1/r. General relativity gives a simple equation (in units where G = c = 1):
d²udφ² = −u + 3Mu²Without the term 3Mu² you get a straight line — this one extra term bends the rays. The program traces rays backward, from the camera toward the scene, and integrates the equation with the 4th-order Runge–Kutta method (RK4). A ray can fall below the horizon (black), cross the disk (glowing gas) or escape to the stars.
The photon sphere and the shadow
At a distance of r = 3GM/c² = 1.5 rs light can travel in a circle — this is the photon sphere (an unstable orbit). Rays passing the hole closer than b = 3√3 GM/c² ≈ 2.6 rs get swallowed, so the shadow looks 2.6 times larger than the horizon. Far from the hole a ray bends by the angle:
α ≈ 4GMc² bThat's twice what Newtonian physics would predict. This is exactly what was measured during the 1919 solar eclipse: starlight near the edge of the Sun's disk bends by 1.75″.
The accretion disk and the Doppler effect
Gas orbits the hole and heats up through friction. The innermost stable orbit (ISCO) lies at 3 rs — that's where the disk begins, and the gas moves at half the speed of light. The side coming toward us is brighter and bluer, the receding side dimmer and redder. On top of that, light loses energy as it climbs out of the gravitational field. Both effects are combined in the factor g:
g = νobsνemitIobs ∝ g⁴ IemitFor gas on a circular orbit g = √(1 − 3M/r) / (1 − Ωλ), where Ω is the angular velocity of the gas and λ — the photon's angular momentum per unit energy (for a very distant camera).
Time and tidal forces
A clock hovering motionless at distance r runs slower than a clock far from the hole:
Δtprobe = Δtfar · 1 − rs/rAt r = 4/3 rs time runs twice as slowly, and at the horizon itself — as seen by a distant observer — it stops. The difference in pull between head and feet (height L) is approximately:
Δa ≈ 2GM · Lr³At the horizon r = rs ∝ M, so Δa falls off as 1/M². The bigger the hole, the gentler it is at its edge.
Gravitational lensing
A mass between us and a distant galaxy bends its light. When everything lies on one line, the image stretches into an Einstein ring with an angular radius of:
θE = 4GMc² · DLSDL DSDL and DS are the distances to the lens and to the source, DLS — between them. The true position of the source β and the image position θ are linked by the lens equation β = θ − θE²/θ, and the brightness grows μ times:
μ = u² + 2u u² + 4,u = β/θEGravitational waves
Two orbiting masses make spacetime ripple and lose energy, so they draw closer and orbit faster and faster. The wave frequency rises until the merger — that's the “chirp.” Its pace depends on the chirp mass:
ℳ = (m1m2)3/5(m1 + m2)1/5τ = 5256(πf)−8/3Gℳc³−5/3For GW150914 ℳ ≈ 28 solar masses, so only about 0.2 s passes from 35 Hz to the merger. After the merger the new hole “rings” like a struck bell (about 250 Hz) and quickly falls silent.
Hawking radiation
In 1974 Stephen Hawking showed that once quantum physics is taken into account, a black hole glows like a very cold body with a temperature of:
T = ħc³8πGMkB ≈ 6·10⁻⁸ K · M☉Mtevap ∝ M³The bigger the hole, the colder it is and the longer it takes to evaporate: a hole with the mass of the Sun would need about 2·10⁶⁷ years — about 10⁵⁷ times the age of the Universe. Hawking radiation has not been observed yet. An open question remains: what happens to the information about what fell into the hole once it evaporates (the information paradox)?
Spinning (Kerr) black holes
Real black holes usually spin (the Kerr solution, 1963). A spinning hole drags spacetime along with it — this is frame dragging. In the ergosphere (drawn here for fast spin, a = 0.9) nothing can stand still: everything must orbit along with the hole, though it is still possible to escape. The horizon is smaller than for a non-spinning hole, and at very fast spin the ISCO moves close to the horizon. Our simulator in mode 1 shows a non-spinning (Schwarzschild) hole.
Model simplifications
- Ray tracing: the black hole does not spin (Schwarzschild); the camera hovers motionless, so we ignore the aberration from its motion.
- The disk is infinitely thin and semi-transparent; its brightness (∝ g⁴) and colors are illustrative — the temperature is rescaled to visible colors. We ignore light travel time; the background lies infinitely far away.
- We compute the ray paths once for each camera distance (a table of about 4,000 orbits) and interpolate.
- Lensing: a point mass in the weak-field approximation; distances and scale are illustrative.
- Time dilation applies to an observer hovering in place. The Hawking evaporation time is computed for photons only.
- S2: a Keplerian orbit (without the small Schwarzschild precession, which was also measured in 2020).
- Gravitational waves: lowest-order approximation (the quadrupole formula) and a schematic merger; amplitude and distances in the drawing are not to scale.
Did you know…?
- April 10, 2019. The Event Horizon Telescope (EHT) released the first image of a black hole's shadow: M87*, about 6.5 billion solar masses, 55 million light-years away. On May 12, 2022 came the image of Sgr A* from the center of our Galaxy (about 4 million solar masses). The EHT is a network of radio telescopes on several continents that works like a single Earth-sized telescope.
- On September 14, 2015, the LIGO detectors recorded gravitational waves for the first time (GW150914) — ripples in spacetime from the collision of two black holes of about 36 and 29 solar masses, 1.3 billion light-years from us. In a fraction of a second they radiated energy equal to about 3 solar masses.
- On the photon sphere (1.5 rs) light can circle the hole. If you stood there looking straight ahead along the orbit, you would see your own back — the light from it would go around the hole and come back to your eyes.
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