Black Hole

A black hole is a place where spacetime is curved so steeply that, past a boundary called the event horizon, every possible path leads inward — even for light. Cross it and you cannot come back or send a signal out.^[1] They form when a massive star collapses at the end of its life, and supermassive ones — millions to billions of times the Sun's mass — sit at the centres of galaxies, including our own. Far from being just theory, one was directly imaged in 2019 by the Event Horizon Telescope, its dark silhouette ringed by glowing gas, and their mergers are the loudest sources of gravitational waves.^[2] At the very centre, general relativity predicts a singularity where its own equations break down.

Beam aim = 0.00 r_s

— escapes— captured┈ inside the horizon┄ photon sphere (r = 1.5 r_s)

A parallel beam of light sweeps past the hole. Rays that pass wide barely bend; the closer they aim, the harder spacetime curves them. Aim within the dashed capture radius (b_c = 3√3·M ≈ 2.6 r_s) and the ray spirals in instead of out — grazing rays can loop the photon sphere before deciding. Once a ray crosses the event horizon, every direction points inward: the dashed red segment shows the path continuing to the singularity, from which no signal returns. Slide to aim the beam across the hole and watch the shadow swallow whichever rays line up with it.

Light itself is not immune. The visualization above traces real light rays through the curved geometry: aim them wide and they merely bend, but aim them within the capture radius and they spiral past the point of no return, crossing the event horizon on their way to the centre.

Deep dive · The Schwarzschild radius

For a non-rotating mass $M$ , the event horizon sits at the Schwarzschild radius,

r_s = \frac{2 G M}{c^2}.

A quick way to see it: set the Newtonian escape velocity $v = \sqrt{2GM/r}$ equal to the speed of light $c$ and solve for $r$ — you get exactly $r_s$ . Below that radius, escaping would mean going faster than light, so nothing does.^[3] It is astonishingly small: crush the Sun to $r_s \approx 3\,\text{km}$ , or the whole Earth to about $9\,\text{mm}$ , and it becomes a black hole.

The full geometry outside the mass is the Schwarzschild metric,

ds^2 = -\left(1 - \frac{r_s}{r}\right)c^2\,dt^2 + \left(1 - \frac{r_s}{r}\right)^{-1} dr^2 + r^2\,d\Omega^2.

Watch the factor $\left(1 - r_s/r\right)$ : as $r \to r_s$ it sends the $dt^2$ term to zero and the $dr^2$ term to infinity — from far away, time appears to freeze at the horizon and distances stretch without limit. Push all the way to $r = 0$ and the curvature diverges: the singularity.

Deep dive · The photon sphere — where light orbits

Just outside the horizon lies a radius where gravity bends light so sharply that a photon can travel in a circle, orbiting the black hole. For a non-rotating hole this photon sphere sits at

r_{\text{ph}} = \tfrac{3}{2}\,r_s = \frac{3 G M}{c^2},

exactly halfway between the horizon and one Schwarzschild radius above it.^[4] These orbits are unstable: a photon nudged infalling spirals down through the horizon, while one nudged outward peels away to infinity. That knife-edge is what you see in the visualization above — rays aimed near the critical impact parameter loop the sphere one or more times before committing to escape or capture.

The photon sphere also sets the black hole's apparent size. Light can escape only if its impact parameter exceeds

b_c = 3\sqrt{3}\,\frac{G M}{c^2} \approx 2.6\,r_s,

so a distant observer sees a dark disk — the shadow — noticeably larger than the horizon itself, since near-grazing rays are swallowed too. This is the ring the Event Horizon Telescope imaged: not the horizon directly, but light skimming the photon sphere and lensed around the hole. A spinning (Kerr) black hole splits it in two — a tighter prograde orbit that co-rotates with the hole and a wider retrograde one against it.