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

Take a clock made of two mirrors facing each other with a pulse of light bouncing between them. Each round trip is one tick. It is a perfectly good clock: regular, countable, and made of nothing but light and geometry.

Now set it moving sideways and watch it go past. The light still bounces between the mirrors, but the mirrors have moved between the bounces, so the light must travel along a diagonal rather than straight up and down. It covers more distance.

And it covers that distance at the same speed, because the speed of light is the same for everybody. More distance at the same speed takes more time. The moving clock ticks slower, and there is nowhere in that chain of reasoning to object.

The triangle #

The whole derivation is one right-angled triangle, and it is worth doing rather than gesturing at, because the result is short enough to be checked in a line.

Let the mirrors be a distance $L$ apart. In the clock’s own frame, a one-way trip takes $t_0 = L/c$.

To the observer watching it fly past at speed $v$, the clock moves sideways by $vt$ during the trip, while the light travels the hypotenuse of a triangle with height $L$ and base $vt$. So

$$(ct)^2 = L^2 + (vt)^2.$$

Solve for $t$:

$$t = \frac{L}{\sqrt{c^2 - v^2}} = \frac{L/c}{\sqrt{1 - v^2/c^2}} = \gamma\, t_0.$$

That is it. The Lorentz factor $\gamma = 1/\sqrt{1 - v^2/c^2}$ is the ratio of the hypotenuse to the height, and it appeared out of Pythagoras applied to a triangle whose sides are fixed by one postulate.

Why every clock has to agree #

The obvious objection is that this proves something about light clocks and nothing about clocks in general. A pendulum, a quartz crystal and a decaying nucleus are not made of bouncing light.

The answer is not a separate calculation for each. It is a consistency argument, and it is much stronger.

Put a light clock and a pendulum clock side by side, synchronised, both at rest. Now set them both moving together. If the light clock slowed and the pendulum did not, the two would drift apart — and an observer travelling with them could compare them, notice the drift, and conclude they were moving.

But there is no such thing as moving. The principle of relativity says the laws of physics are the same in every uniformly moving frame, so no experiment performed inside a sealed laboratory can reveal its velocity. Two clocks drifting apart would be exactly such an experiment.

So every clock must slow by exactly the same factor, including the biological and chemical processes that constitute ageing. Not because time dilation acts on each mechanism separately, but because time itself is what has been rescaled, and the mechanisms are only reading it.

The argument is a good example of a move that recurs throughout physics: rather than calculating the effect on each of a hundred mechanisms, forbid the outcome that would let an observer detect something forbidden, and let that forbid the hundred cases at once. The proof that no engine beats the Carnot ceiling has exactly this shape, and so does the argument that charge cannot accumulate inside a conductor. A statement about what cannot be observed does far more work than a statement about what happens.

This is the point at which the effect stops being about clocks. The light clock is a device for making visible something about the structure of spacetime, and once the argument is made, the device can be thrown away.

The disagreement is symmetric #

The result reads as though moving clocks are slow, which suggests one of the two observers is right. Neither is.

From the moving clock’s own frame, it is at rest and the other clock is moving — so the other clock is running slow, by the same factor. Both observers are correct, and both are making a statement about the other’s clock rather than about time in general.

This sounds like a contradiction and is not, and the resolution is not a subtlety about measurement. It is that the two observers are not comparing the same things. Comparing a moving clock against stationary clocks requires two stationary clocks at different places, synchronised — and whether two separated clocks are synchronised is exactly what observers in relative motion disagree about.

Time dilation and the relativity of simultaneity are not two effects. They are one effect described in two ways, and any account of the first that omits the second will eventually produce a paradox.

Seeing both pictures is worth the duplication. The diagram makes the result structural — a consequence of how the axes are related — while the light clock makes it mechanical, with an object whose ticking can be watched. Neither is more correct. The diagram generalises better and the clock convinces better, and a reader who has only one of them tends to believe the result for the wrong reason.

The evidence #

The effect is confirmed to high precision, and the confirmations are not delicate laboratory curiosities.

Muons. Cosmic rays striking the upper atmosphere — themselves the products of collisions at enormous energies — produce muons about fifteen kilometres up. A muon lives 2.2 microseconds on average, and even at light speed that permits only 660 metres of travel. Almost none should reach the ground; large numbers do. At the speeds involved, $\gamma$ is around 20, and 660 metres becomes thirteen kilometres. The muon’s own account is different and equally valid — it lives its usual 2.2 microseconds and the atmosphere, contracted, is only 750 metres thick.

Particle accelerators. Unstable particles in storage rings live longer by precisely $\gamma$, measured routinely, at values of $\gamma$ in the thousands. This is not a test that anybody bothers to run any more; it is a design constraint that has to be right for the machines to work at all.

Atomic clocks on aircraft. Hafele and Keating flew caesium clocks around the world in 1971 and compared them with clocks left behind. The differences were tens of nanoseconds and matched prediction — which had to combine the speed effect with the gravitational one, in opposite directions.

Satellite navigation. GPS satellites orbit at about 3.9 km/s, which loses their clocks 7 microseconds a day, and sit high in a weaker gravitational field, which gains them 45. The net 38 microseconds a day is corrected for in the satellites’ clock rates before launch. Uncorrected, positions would drift by about ten kilometres a day. The system is a continuously running relativity experiment that a great many people depend on without knowing it.

What the picture cannot show #

The flatness of that curve is a fact about the physics and a warning about the drawing.

The diagram exaggerates. At any speed for which the geometry is visible on a page, the situation is unimaginably far from ordinary experience. At the speed of a passenger aircraft, the diagonal in the light-clock figure would differ from the vertical by about one part in $10^{12}$ — a deviation smaller than the width of an atom over the width of the page.

The picture is a mixed frame. The two clocks are drawn side by side as though a single observer could see both behaving that way. That observer is stationary, so the “at rest” clock is the honest one and the “moving” one is being described from outside. There is no viewpoint from which both drawings are simultaneously first-person.

It shows one tick. Time dilation accumulates, and the effect that matters is always a total elapsed time along a path. The right quantity is proper time — the time measured by a clock carried along a worldline — and it depends on the whole route rather than only its endpoints.

It assumes constant velocity. Nothing in the derivation permits the clock to speed up or slow down. Acceleration is where the symmetry breaks, and where the twin paradox stops being paradoxical: two twins who separate and reunite have travelled different worldlines between the same two events, and the one who accelerated has aged less. The asymmetry is geometric, not psychological — one path is longer in proper time than the other, and there is no more contradiction in that than in two roads between the same towns having different lengths.

Where the analogy with distance inverts, and this is the memorable part: in spacetime the straight path is the one with the most proper time. The twin who stays put ages more. A geodesic maximises rather than minimises, which follows directly from the minus sign in the invariant interval.

Where the model stops #

Special relativity only. The derivation assumes flat spacetime with no gravity. Clocks also run slow lower in a gravitational field, by a different amount from a different cause, and any real measurement — including every one listed above — must combine both.

Uniform motion. Covered above, and the reason the twin case needs more than this argument.

Rigid mirrors at a fixed separation. The clock’s own length is assumed constant as it accelerates to speed. A genuinely rigid object is not permitted by relativity at all, since rigidity would transmit influence instantaneously, so this is an idealisation with real content behind it.

A classical light pulse. The pulse is treated as a point travelling on a definite path. It is a wave, and a finite pulse contains a spread of frequencies; nothing in the argument breaks, but the picture is a simplification of an object with structure — and a wave is not the kind of thing that has a single trajectory to draw.

The ladder from here #

Later rungs: length contraction from the same triangle rotated. Proper time as the length of a worldline, and the twin paradox drawn rather than argued. The relativistic Doppler effect, which combines dilation with the ordinary wave shift. Velocity addition and why light speed cannot be exceeded by combining speeds. Relativistic momentum and energy. Gravitational time dilation and the equivalence principle. The Pound–Rebka experiment, which measured a frequency shift over twenty-two metres of a Harvard tower. And the relativity of simultaneity taken as primary, which is the shortest route to every result here.

Einstein’s 1905 paper contains no light clock. The derivation everybody now learns first was constructed afterwards, as a way of showing that the conclusion needed nothing but a triangle.