Notes on the chart
Everything moves by the Schwarzschild solution of Einstein’s equations: spacetime outside a non-rotating, uncharged mass. Distances are in Schwarzschild radii rₛ (the radius of the horizon) and times in rₛ/c. The units switch converts them for a black hole of one solar mass (rₛ = 2.95 km) or for Sagittarius A* (4.3 million solar masses, rₛ = 12.7 million km).
A clock held still at r ticks at √(1 − 1/r) of the rate of a clock far away. Nothing can be held still at or inside the horizon.
Light rays are integrated exactly from the geodesic equation. A laser’s aim is the direction measured by an observer held still where it stands. Light whose impact parameter is below 3√3/2 ≈ 2.60 rₛ is captured; at 1.5 rₛ it can circle the hole. Pulses move along each ray by far-away time, so they appear to slow down near the hole.
A probe let go from rest falls on the exact cycloid solution: it crosses the horizon after a finite time on its own clock. Orbits are integrated with their conserved energy and angular momentum; circular orbits are stable only outside 3 rₛ, and nearly circular ones precess by 2π / √(1 − 3/r) per lap.
The map is a history of everything on the chart: distance from the centre across, time up. The time is Eddington–Finkelstein time t̃ = t + ln|r − 1|: it ticks like a far-away clock but stays finite at the horizon, so a falling probe can be followed through it. Something held still is a vertical line; something falling leans left as it climbs.
Each cone shows where a flash of light at that point could go next: the left arm is light moving inwards, the right arm light moving outwards, and anything with mass must travel between them. Far out the cone is symmetric; nearer the hole the outward arm steepens; at the horizon it stands upright (outgoing light hovers there); inside, both arms lean inwards. By default the time axis is 1:1, so light far away climbs at 45°; “Time squeezed” fits more history in but flattens the cones.
The river view draws the same spacetime in Gullstrand–Painlevé form: space is flat, and flows inwards at v = √(rₛ/r) c, the speed of a raindrop falling from rest far away, as measured by an observer held still where it passes. Light moves at c relative to that flow in every direction. A clock held still moves through the flow at speed v, so it runs slow by √(1 − v²/c²) = √(1 − rₛ/r): the gravitational slowing is the special-relativity slowing of swimming against the river. Flashes are drawn from 48 exactly traced rays; the green dot marks where the river has carried the flash’s centre. The dots of the river are drawn for illustration (a sample of raindrops), but they move at the exact raindrop speed.
Drag a part to move it; drag a laser’s round handle to aim it. Arrows nudge, [ and ] turn a laser, Delete removes, M switches to the map, 1–8 choose a plate.
Drawn for clarity: the size of clocks, probes and markers, and the speed of play. Rotation, charge, tides and the hole’s growth are left out; near a solar-mass hole a real probe would be torn apart before the horizon (not near Sagittarius A*).
Further reading: Taylor & Wheeler, Exploring Black Holes · Misner, Thorne & Wheeler, Gravitation, §31
Theoria · Curved spacetime 0.6 · Chart table