Einstein replaced the force of gravity with geometry: mass curves spacetime and everything, light included, follows the straightest path it can. Launch particles and light rays around a black hole and see where Newton's answer stops working.
Drag on the sheet to launch a particle: press where it starts and pull in the direction it should move. Longer drags launch faster.
A non-rotating black hole of mass M sits at the centre. The black disc is its event horizon at the Schwarzschild radius rs = 2GM/c². The funnel is Flamm's paraboloid, the standard embedding diagram of how space around the hole is curved, with its depth scaled to fit.
Einstein's particles follow the exact Schwarzschild orbit equation, d²u/dφ² + u = GM/h² + 3GMu²/c² with u = 1/r. The last term is the whole difference from Newton. Light obeys the same equation without the GM/h² term.
Nothing pulls on the planets. Each one moves as straight as it can through curved spacetime, and the curve shows up as an orbit. Close to the hole the geometry gets so steep that no stable orbit exists below 3 rs, and at 1.5 rs even light, the fastest thing there is, can only circle.
Einstein explained Mercury's orbit with this in 1915. Eddington's eclipse expedition measured starlight bending by the Sun in 1919, twice the amount a Newtonian calculation gives.
The effects in this demo are tiny near Earth but they are measured every day, and engineers and astronomers have to build them in.
GPS satellites orbit 20,200 km up, where spacetime is less curved, so their clocks gain about 45 microseconds a day on clocks on the ground. Their orbital speed slows them by about 7, for a net 38 µs a day. Left uncorrected, positions would drift by around 10 km a day, so the satellite clocks are set slightly slow before launch.
In the demo: the funnel is shallower far out; clocks there tick faster.Mercury's closest point to the Sun shifts by 574 arcseconds per century. Pulls from other planets explain all but about 43. Einstein's 1915 calculation gave exactly those missing 43 arcseconds, the first evidence that general relativity was right.
In the demo: Precessing shows the same effect, made huge by orbiting near a black hole.Galaxy clusters bend the light of galaxies behind them into arcs and distorted shapes. Measuring that distortion maps all the mass in the cluster, visible or not. Lensing maps such as the Bullet Cluster show most of the mass sits where there is no glowing matter, strong evidence for dark matter.
In the demo: light at large b bends by 4GM/(c²b), so bending reveals mass.When a star passes almost exactly in front of a more distant one, its gravity briefly focuses the background light and it brightens for days or weeks. A planet around the foreground star adds a short extra blip. Surveys like OGLE and KMTNet have found well over a hundred planets this way, including cold, distant ones other methods miss.
In the demo: the ray family converges behind the mass, like a lens.The Event Horizon Telescope linked radio dishes across the planet into one Earth-sized telescope. Its images of M87* in 2019 and Sagittarius A* in 2022 show a bright ring around a dark centre. The ring's size comes from light bent around the photon sphere and matches general relativity's prediction.
In the demo: rays with b near 2.6 rs orbit the photon sphere.Radio signals that pass close to the Sun arrive late, because the curved path through warped spacetime takes longer. Irwin Shapiro predicted it in 1964 and measured it by bouncing radar off Venus and Mercury. In 2003 the Cassini spacecraft's radio link matched Einstein's prediction to within about 0.002%.
In the demo: rays near the mass take the long way round.