Illustrated Physics
physics, drawn
Concept

The twin paradox — where it appears

The apparent contradiction of two travellers each finding the other's clock slow, resolved by proper time being a property of a path. The symmetry is broken by one traveller changing frames, and the difference in elapsed time is a geometric fact rather than an effect of the acceleration itself.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

A light clock at β = 0.6. The same clock at rest and moving. Light covers the hypotenuse rather than the height, and since its speed is the same for both observers, the moving clock must take longer to tick.

The clock that has to slow, and why no clock can refuse

One constant speed and one right-angled triangle force a moving clock to tick slower. The argument is Pythagoras, which is what makes it inescapable rather than merely surprising.

relativity · Time dilation A spacetime diagram at β = 0.6. Position across, time up, in units where light travels at 45°. The shaded wedges are the future and past reachable by light; the tilted axes belong to an observer moving at 0.6 of the speed of light.

The twin who comes back younger

If motion slows a clock, and motion is relative, each twin should find the other younger — and yet when they meet, one of them has aged less. The asymmetry is not in the speed and not in the acceleration; it is in which worldline is straight.

relativity · Time dilation What puts a scale on a tilted axis. A spacetime diagram with a second observer's axes at β = 0.6. The curves are the sets of events at a fixed interval from the origin — c²t² − x² = s², one branch each for s = 0.5, s = 1, s = 1.5 — and the whole point of them is where they cross. A unit of the moving observer's time is wherever the s = 1 curve meets the tilted time axis, and on the page that point is 1.250 times as far from the origin as the stationary observer's own unit. Without the hyperbolae the tilted axes carry no scale at all, and every argument about which of two clocks is behind is unreadable off the diagram. Each drawn crossing reads back as its own interval to 0.0e+0.

The quantity nobody argues about

Relativity takes away the length of a rod and the duration of an event and hands back exactly one thing in their place. Its hyperbolae are what put a scale on the tilted axes of a spacetime diagram — without which the diagram is a picture with no units on it.

relativity · Spacetime diagram One factor, defined by an experiment rather than by a transformation. A observer A stays at x = 0 and flashes a light every 1 second by their own clock. B recedes at 0.6c. The flashes are the diagonal lines; where each meets B's worldline is where B receives it. B's clock reads a longer gap between arrivals than A's read between departures, by the factor k = 2.0000, and it is the same factor between every consecutive pair — measured here off the drawn meetings rather than assumed. That single number is the whole apparatus. Nobody has written down a coordinate transformation, chosen a convention for distant simultaneity, or drawn a tilted axis; the only thing used is that light travels on the diagonals and that neither observer is special, so B's flashes reach A stretched by the same k. From it: γ = (k + 1/k)/2 = 1.2500, and β = (k² − 1)/(k² + 1) = 0.6000.

Everything from an exchange of pulses

Send a flash every second and ask how often the far observer receives them. That one measured ratio generates time dilation, the composition of velocities and the twin result, with no coordinate transformation written down anywhere and no convention chosen about what "at the same time" means far away.

relativity · Doppler The turnaround, made gentler and gentler, and the difference that does not move. A round trip to a star 4 light-years away at 0.6c, with the turnaround done at nine different accelerations from a tenth of a gravity to a thousand. The upper curve is the age difference between the twins and the lower one is how much of that difference the turnaround itself contributes. At 0.1 g the turn accounts for 51 per cent of it; at 1000 g it accounts for 0.00 per cent, and it keeps falling. The total does not follow it down: it tends to 2.67 years, which is what the instantaneous-turnaround cartoon gives. So the acceleration is not what makes the twins differ. It is what makes one twin's path the bent one, and a bent path through spacetime is shorter for the same reason a bent path on a map is longer — but the amount is in the legs, not in the corner, and the corner's contribution can be made as small as anyone likes without the difference going away.

The clock that does not feel the turn

Proper time is the integral of dt over gamma, which presumes that a clock's rate depends on its speed and on nothing else — not on its acceleration, not on how long it has been accelerating. That is an assumption about clocks rather than a theorem about spacetime, and the twin result is empty without it.

relativity · Accelerated frames The paths in space: orbits of one period, and a throw straight up. The same free falls drawn in space around the Earth, which is the filled disc. All start at the marked point 2 Earth radii from the centre. The circle is the circular orbit. The ellipses, of eccentricity 0.2 and 0.4, have the same period, so they come back to the start at the same moment. The straight line is the thrown clock's path: straight up to 4.46 Earth radii and back down the same line, arriving as the orbits complete one revolution. The Earth's rotation is ignored and it is treated as a point mass for the paths that pass close to it.

The orbit that ages less than a throw

A clock in orbit and a clock thrown straight up leave the same point at the same moment and meet there again one period later. Both fall freely the whole way, so both follow paths of stationary proper time — and the thrown clock comes back 4.1 microseconds older. Even a clock held still by a rocket, which is not falling at all, beats the orbit. Free fall picks out a path that is stationary, not one that is longest.

relativity · Time dilation

Named alongside it

The objects these essays reach for when they reach for this one.

Proper timeTime dilationThe Lorentz factorSimultaneityInvariant intervalAccelerationCausalityClock hypothesisConjugate pointDoppler effectExperimental boundFree fall

All concepts