Relativity,
in three dimensions.

Both descriptions are correct. The ball’s path depends on who is watching, and nothing you do inside the train can tell you whether it is moving or parked. Galileo made this argument about a ship’s cabin in 1632.

That is the principle of relativity: the laws of physics are the same for everyone who moves at a steady speed. There is no special state of “at rest”, only motion relative to something else.

Try it. Set the speed to zero and switch views: they agree. Raise the speed and switch again. Only the path changes, never the physics.

For centuries that settled it. Then came light.

Adding speeds says a pulse fired forward from a ship moving at 0.8 c should travel at 1.8 c. That is the red ghost. Nature disagrees: the pulse always travels at exactly c, however fast the ship goes.

This has been tested directly. In 1964 a team at CERN measured light from particles that were themselves moving at 99.975% of the speed of light. The light still travelled at c, to within about one part in ten thousand.

In 1905 Einstein built special relativity on two statements. The laws of physics are the same in every steady frame, which is chapter 1. And light travels at the same speed for everyone, which is this chapter.

The catch. The station sees the pulse pull away from the ship at only (1 − v/c) times c. The ship’s crew sees it leave at exactly c. Both measure honestly, so their clocks and rulers must disagree. Next: how.

Standing still, the clock ticks every 2L/c seconds: the photon climbs straight up, drops straight down, and repeats.

From the station, the moving clock is different. While the photon crosses from one mirror to the other, the clock has slid forward, so the photon’s path is a diagonal: up and forward, then down and forward. That is a longer trip than L, and light can’t speed up to make up the difference, because everyone measures the same c. So each tick takes longer. To the station, the moving clock runs slow.

Each half-tick is a right triangle: the clock’s height L, the distance vΔt/2 it slides, and the photon’s path cΔt/2. Pythagoras gives the rest.

(cΔt/2)² = L² + (vΔt/2)²

Solve for the time between ticks and it is the stationary tick stretched by a factor called the Lorentz factor, γ.

Δt = γ Δt₀,  γ = 1 / √(1 − v²/c²)

Try it. At 0.8 c, γ is 5/3: the moving clock ticks three times for every five ticks of the stationary one. Above 0.95 c the zigzag grows too wide to fit the scene, but the graph carries on.

Show the numbers

Speed	Slowdown, γ	Moving-clock ticks per 10 stationary
0.10 c	1.01	9.95
0.50 c	1.15	8.66
0.80 c	1.67	6.00
0.90 c	2.29	4.36
0.95 c	3.20	3.12
0.99 c	7.09	1.41
0.999 c	22.4	0.45

γ is 1 when nothing moves and grows slowly at first: 1.15 at half the speed of light, 2 at 87%, 7 at 99%, and without limit as the speed approaches c.

This isn’t a property of mirrors. Every clock and process aboard, from a heartbeat to a decaying particle, has to slow by the same factor. Otherwise the crew could tell they were moving, which breaks the principle of relativity from chapter 1.

Motion is relative, so the crew says the same of you: your clock is the slow one. Both are right, because they disagree about which events in different places happen at the same time. Chapter 4 shows what breaks the tie.

The ship’s clock runs slow by the factor γ from the last chapter, so a trip that takes 2d/v years by Earth’s calendar takes only 2d/(γv) aboard. At 0.9 c, a round trip to Sirius, 8.6 light-years away, takes 19.1 years on Earth and just 8.3 for the traveler.

The traveler tells it differently. From the ship, the road ahead is squeezed to 1/γ of its length: Sirius is only 3.7 light-years away, so the trip is shorter. Two explanations, one outcome.

Why isn’t it symmetric, with each twin seeing the other age slowly? Because only one of them turns around. The traveler switches from a frame heading out to a frame heading home, and the stay-at-home twin never changes frame.

Try it. Drag the speed toward 0.99 c and watch the green bar fall behind. Drag the progress slider to check any moment of the trip.

Clocks have made this trip too, at everyday speeds. In 1971 Hafele and Keating flew atomic clocks around the world on airliners and found them out of step with clocks on the ground, by tens to hundreds of nanoseconds, just as relativity predicted.

Light covers about 660 metres in 2.2 microseconds, so a muon that moves almost that fast should typically decay after roughly 660 metres. Surviving all 15 kilometres would be a one-in-ten-billion chance. Choose Ignore relativity and see: none of the 300 muons reaches the ground.

Switch it back on. From the ground, a muon’s clock runs slow by γ, so its life is stretched by the same factor and about one in ten arrive at 0.995 c. That is what detectors measure. In 1963 Frisch and Smith counted muons on a mountaintop and at sea level and found that their decay had slowed by a factor of about 9, close to the 10 predicted for their speed.

The muon’s own view is different and just as valid. Its clock ticks normally, but the atmosphere rushes toward it, squeezed to 1/γ of its height: about 1.5 kilometres instead of 15. That is the small column on the right. There is time to cross it.

Try it. Drag the speed toward 0.9999 c and watch the right-hand column shrink while more muons arrive. Two explanations, one outcome: everyone agrees the muon arrives.

Einstein’s general theory of relativity, published in 1915, describes gravity as the curving of space and time by mass. The sheet in the scene is the familiar picture of that curve, an analogy rather than a literal surface. A clock deeper in the well ticks more slowly than one far outside it.

rate = √(1 − r_s/r)

Here r is the distance from the mass and r_s = 2GM/c² is its Schwarzschild radius: about 3 kilometres for the Sun and 9 millimetres for Earth. At r = r_s, the horizon of a black hole, the rate reaches zero.

Try it. Slide the probe from 12 down to 1.1 Schwarzschild radii and watch its hand slow. Near the horizon it ticks at less than a third of the far clock’s pace.

On Earth the effect is tiny but real. A clock on the ground runs slow by about 7 parts in ten billion compared with one far away, roughly 22 milliseconds a year. In 2010, physicists at NIST compared two atomic clocks and measured the difference made by raising one of them just 33 centimetres.

Your phone depends on it. GPS satellites orbit about 20,200 kilometres up, where gravity is weaker, so their clocks gain about 46 microseconds a day. Their speed makes them lose about 7. The net gain, roughly 38 microseconds a day, would put positions off by about 11 kilometres every day if the system did not correct for it.