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 the mirrors are side by side #

The clock in the figures is oriented across the direction of motion, and that is not a drawing convenience. It is the choice that makes the derivation self-contained, and turning the clock through ninety degrees shows why.

The transverse arrangement works because the separation $L$ between the mirrors is unaffected by the motion. That is not assumed; it follows from a symmetry argument that has to be made once. If transverse lengths changed with speed, then of two observers passing each other, each would say the other was narrower — and a ring flying over a rod would both fit and fail to fit, which is a disagreement about a local coincidence rather than about a coordinate. No such disagreement is permitted, so transverse lengths are unchanged and the triangle’s height is $L$ in both frames.

Now lay the clock along the direction of travel instead. The light chases the receding front mirror, taking $L/(c-v)$, and returns to the approaching back mirror in $L/(c+v)$. The round trip is $2Lc/(c^2-v^2)$, which is $\gamma^2$ times the resting tick — too slow by a whole extra factor of $\gamma$.

The two clocks must agree, because they are two clocks, and this page’s next section is about why no clock may disagree with another. So the longitudinal clock’s length cannot be $L$: it has to be $L/\gamma$, and the factor is recovered exactly.

That is length contraction, obtained here as a requirement rather than as a separate result. And the two orientations together are the Michelson–Morley apparatus, which is two light clocks at right angles asked whether they keep the same time. They do, which is what the null result says.

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.

The disagreement is symmetric, and what makes that consistent rather than contradictory is simultaneity. Two events that happen at the same moment for one observer do not for the other — so when each says the other’s clock runs slow, they are comparing against different pairs of events. There is no shared “now” for the two claims to contradict each other in.

Put as a spacetime diagram the same statement becomes geometry. The moving clock’s ticks are marked along its own tilted time axis, and projecting them onto the stationary axis is exactly what “comparing the two clocks” means — a projection, which shortens things, and which each observer performs onto their own axis. Two projections in opposite directions is not a contradiction; it is what a rotation looks like when both parties do it.

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.

What the constant speed costs #

Holding one quantity fixed for every observer is not free. Something has to absorb the difference, and the derivation above shows what does: the triangle keeps its shape only because the time is allowed to change. That is the pattern of the whole theory. Insisting that $c$ is the same in every frame is paid for by giving up the frame-independence of nearly everything else.

The bill, itemised: durations become frame-dependent, as above. Lengths do too. Simultaneity stops being a property of a pair of events and becomes a property of a pair of events and a frame. Velocities no longer add. Momentum is no longer $mv$. Kinetic energy is no longer $\tfrac12 mv^2$. The ordering in time of two events with no causal connection stops being a fact and becomes a convention. Every one of those was regarded as self-evident in 1900, and each is surrendered to keep a single number constant.

The trade is worth it because it buys something specific: one set of laws that works in every frame, rather than one set that works in a preferred frame and a family of corrections everywhere else. It is the same kind of bargain as choosing axes to suit the constraint rather than the room, taken to its limit — accept an awkward-looking description in exchange for the disappearance of an entire category of special case.

There is an engineering cost too, and it is paid daily. The most expensive consequence is that the flat part of the $\gamma$ curve turns vertical: energy buys speed generously at first and then almost not at all. A proton in the Large Hadron Collider carries about 6.5 TeV, which is $\gamma \approx 6{,}900$ — and its speed is $c$ short by roughly three metres per second. Doubling the energy of the machine, at a cost of billions, changes that shortfall to a bit under one metre per second and changes the speed not at all measurably. What the energy actually buys is $\gamma$: more time in the particle’s own frame, more mass-energy available in a collision, and a longer laboratory lifetime for whatever is produced. Accelerator physics is arithmetic done in $\gamma$ rather than in $v$ for the simple reason that $v$ stopped being informative around the first bend.

The muon storage ring at Fermilab runs at $\gamma = 29.3$, which stretches the muon’s 2.2 microseconds to 64.4, and the entire experiment depends on that number being right to better than a part in $10^{5}$. Time dilation stopped being a result to be tested and became a piece of equipment.

The measurement that forced it #

The light clock is a modern teaching device, and the history ran the other way round: the geometry was reverse-engineered from an experiment that stubbornly found nothing.

Michelson and Morley set out in 1887 to measure the Earth’s motion through the luminiferous ether by splitting a beam of light, sending the halves along two perpendicular arms, and recombining them. If light moved at a fixed speed relative to the ether, the arm lying along the Earth’s motion would be traversed at a different effective speed from the arm across it, and rotating the apparatus would slide the interference fringes. The predicted shift was about four-tenths of a fringe, which their interferometer could see forty times over.

The fringes did not move. The result was so unwelcome that it was repeated for decades, at higher precision, at different times of year, up mountains, and in the 1930s with the apparatus in a vacuum, in case the ether were being dragged along. It has never moved.

FitzGerald and Lorentz proposed a rescue: objects moving through the ether contract along the direction of motion by exactly the factor needed to cancel the effect. This works, and Lorentz went on to derive the full transformation — the equations now named after him — while continuing to regard the ether as real and the contraction as a dynamical effect of motion through it. The mathematics was complete before the interpretation was.

Einstein’s 1905 move was to stop rescuing. Take the null result at face value: the speed of light is the same for everybody, full stop, and the ether is not undetectable but absent. Everything Lorentz had derived follows, and follows from two sentences rather than from a theory of how matter deforms. The paper contains no light clock and no diagram. The triangle in these figures was constructed later, by people who wanted to show that a result of that magnitude needed nothing but Pythagoras.

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.

Ten seconds to cross the galaxy #

The largest $\gamma$ ever recorded belongs to a single proton, and its account of its own journey is worth writing down because it is the effect on this page taken as far as nature has been observed to take it.

On the night of 15 October 1991, the Fly’s Eye detector in Utah recorded a cosmic-ray air shower whose primary particle carried about $3\times10^{20}$ electronvolts — some fifty joules, which is a hard-hit baseball’s worth of kinetic energy in one proton. Against a rest energy of 0.94 GeV, that is a Lorentz factor of roughly $3\times10^{11}$.

Take that literally and run the triangle. The particle’s speed falls short of $c$ by about four parts in $10^{24}$: in a hundred-thousand-year race across the Milky Way against a photon starting beside it, the proton would finish some ten picoseconds behind. And by its own clock the crossing takes a hundred thousand years divided by $3\times10^{11}$, which is about nine seconds.

Nothing in that is a different physics from the light clock at the top of this page. It is the same hypotenuse, at a ratio the figures cannot draw: at this $\gamma$ the diagonal is three hundred billion times the height, so any honest picture of it is a vertical line and a horizontal one.

It is also the clearest statement of what the factor does and does not do. The proton is not going any faster than a muon at $\gamma$ of 20 in any way an odometer could tell — the two differ in speed by parts in $10^{22}$. What differs is how much of the journey their own clocks record, and that is the quantity the whole page has been about.

What the picture cannot show #

What the picture cannot show is how flat the curve is. Nothing happens until it does: the Lorentz factor is indistinguishable from one across the whole range of everyday and even orbital speeds, and rises without limit only in the last stretch. That is why the effect went unnoticed for two centuries of careful mechanics, and why it is unavoidable in a particle accelerator.

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 #

The next rung is length contraction, which is this same triangle rotated and turns out to be a statement about simultaneity wearing a disguise. After it: 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.

The muon’s own account of its journey — the same fifteen kilometres, arrived at by a contracted atmosphere rather than a stretched lifetime — is the shortest demonstration that these are one effect and not two, and it is the first thing the next rung has to draw.