Physics · Module 19
Two harmless-looking statements, each supported by every experiment ever done, cannot both be true unless time is not the same for everyone. Einstein kept both statements and gave up universal time.
This module has been coming since Module 15. Let us collect the three things that do not fit.
Thread one: the rod on the rails. In Module 17 we found the EMF of a rod sliding along rails in a magnetic field, twice, by two arguments that shared not a single step. One used the magnetic force qvB — q the charge, v, the speed, B the field — on the electrons in the moving rod. The other used the growing area of the circuit and Faraday's flux rule, and never mentioned a force on a charge at all. Both gave BLv. Feynman flagged this as unique in physics: one experimental fact, two unrelated explanations, always agreeing.
Thread two: the magnet and the coil. Take Bench 16A and ask a question Module 16 did not. Hold the coil still and push the magnet in, and you say: the flux changed, Faraday's law made an electric field, the electrons were pushed. Now hold the magnet still and move the coil instead. Your explanation changes completely. There is no changing flux where the coil used to be. Instead, the electrons in the moving wire feel qvB. Two different mechanisms. And the galvanometer needle reads exactly the same in both cases, always, to any precision anyone has ever measured. It does not know or care which of you is moving.
Einstein opened his 1905 paper with precisely this observation. His first sentence says that Maxwell's electrodynamics, applied to moving bodies, leads to asymmetries that do not appear in the phenomena themselves; his second gives the example — "the reciprocal electrodynamic action of a magnet and a conductor". He found it intolerable that nature should need two separate explanations for one result. The only thing dividing them is the question "which one is really moving?" — and no experiment can answer that.
Thread three: Maxwell's c. The equations give one speed for light, built out of µ0, the permeability of free space, and ε0, the permittivity — the two you met in Module 18 — and there is nothing in them to measure that speed relative to. The nineteenth century's answer was the aether: a medium filling space, with c being the speed relative to it. Michelson and Morley went looking for the Earth's motion through that medium in 1887, with an instrument sensitive enough to find it several times over.
They found nothing. Then they improved the apparatus and found nothing more precisely. Others repeated it at different seasons, with the Earth moving in opposite directions round the Sun. Nothing, every time.
Three threads, one knot. In 1905 Einstein untied it by refusing to look for a mechanism and instead asking what would have to be true.
A postulate is a statement you decide to assume, without proving it, and then judge entirely by what follows from it. Einstein's whole theory rests on two postulates. Read them carefully. They sound mild. The second one is not mild at all — it quietly wrecks something you have believed since Module 1.
Sit with how strange the second one is. Fire a bullet forward from a moving train and the ground observer sees bullet-speed plus train-speed. That is not a theory; it is arithmetic. It has been true since Module 1.
Now switch on a torch on that train. According to postulate 2, the person on the platform measures the light at c. Not c plus the train's speed — c. And if you now chase the beam in a rocket at 0.9c, you still measure it receding from you at c, not at 0.1c.
That cannot be right. And yet it is what Maxwell's equations demand, what Michelson and Morley found, and what every experiment since has confirmed to preposterous precision. Modern versions of the Michelson–Morley experiment would detect a change in light-speed of one part in 1017.
Here is the argument that gets you from the postulates to a number, and it needs nothing beyond the theorem you learnt in class 8.
Build a clock out of light. Two mirrors facing each other, a distance L apart. A pulse of light bounces between them. Every time it makes the round trip, the clock ticks. The tick interval is t0 = 2L/c. It is a perfectly good clock. In fact it is the most honest clock there is: it measures time using nothing but the one speed everybody agrees on.
Now watch an identical clock go past you at speed v.
One thing to be clear about before answering the next question. "Measures" here does not mean "sees". A clock racing away from you also looks slow because each tick's light has further to travel; one coming toward you looks fast. That is the Doppler effect, and it is not what we are talking about. Time dilation is what a row of synchronised clocks laid along the track records as the moving clock passes each one — no light-travel delay, just counting.
Two identical light clocks. One sits on your desk; the other flies past at 0.8c. You count the ticks of both.
What do you find — and, more importantly, what does the person flying past find when they compare the two?
The symmetry is the part people refuse to believe, and it is not optional: it is postulate 1. From the flying observer's point of view they are perfectly still and you are the one sliding past at 0.8c, so it is your light pulse that has to travel diagonally, and your clock that runs slow. Both of you are right, and there is no contradiction. To catch either of you out, the two clocks would have to be brought together twice — once to set them and once to compare. That never happens here: you pass each other once and separate for ever. Bringing them back together needs one of you to turn around, and turning around means accelerating. Acceleration breaks the symmetry: postulate 1 only covers non-accelerating frames, so the twin who turns round is the odd one out — and they really do come home younger.
Let it run for a minute and the counted ratio is already within a decimal place of γ — run it longer and the last digit stabilises. Nothing in the code computes the tick rate from a formula. The pulse is moved at a fixed speed along whatever path it has to take. A tick is counted when it arrives. The slowing is a geometrical consequence of a longer path at a fixed speed, and nothing else.
In 1971 Hafele and Keating flew four caesium atomic clocks around the world on scheduled airliners, eastward and westward, and compared them with identical clocks left at the US Naval Observatory. The clocks disagreed by the predicted amount in each case — the eastward ones slow, the westward ones actually fast, because the prediction needed a gravitational contribution as well as a speed one, and getting both signs right was the real test. Particle accelerators do the experiment continuously and much more brutally: unstable particles kept going round at 0.9994c inside a ring of magnets live about 29 times longer than the same particles at rest — the acceleration that keeps them on the ring does not affect the rate, because what sets a clock's pace at each moment is its speed, not its acceleration. The effect is not subtle in that regime; it is the difference between the particle surviving the trip and not.
Time dilation gets the headlines. The relativity of simultaneity is stranger, and it is the one that actually resolves the paradoxes.
Here is Einstein's own picture. A train carriage, moving. A lamp exactly at its centre. At some moment the lamp flashes, and the light spreads out both ways.
Ask the passenger. The lamp is at the middle of the carriage. The light goes both ways at c. The walls are equally far away and are not moving relative to them. So the light reaches the front wall and the back wall at the same instant. Obviously.
Ask someone standing on the platform. The light left the middle of the carriage and goes both ways at c — the same c, by postulate 2, regardless of the fact that the lamp was moving when it flashed. But in the time the light is travelling, the carriage has moved forward. The back wall is rushing towards the beam that is heading backwards; the front wall is running away from the beam heading forwards. So the light hits the back first, by a clear margin. Not at the same instant at all.
The passenger says the two events — light hitting the front wall, light hitting the back wall — happened simultaneously. The platform observer says the back one happened first.
Which one is mistaken?
Neither is mistaken, and this is not a wording trick. "Now, over there" is not a well-defined idea in physics — it never was; we simply never moved fast enough to notice. Two events at different places have no universal time order unless a light signal could have connected them. One thing is not up for grabs, and the Deeper box says what it is.
This is what unties every knot in relativity. Consider the classic worry: if each of us sees the other's clock running slow, surely someone must be wrong? No — because to compare two separated clocks you have to decide what "at the same moment" means, and that is exactly what the two of you disagree about. The disagreement about clock rates and the disagreement about simultaneity are two views of one thing, and they cancel perfectly.
Not everything is up for grabs. Suppose event A could have caused event B — meaning a signal at c or slower could have got from A to B. Then every observer agrees that A came first. Relativity only permits disagreement about the order of two events when no signal could have connected them — and in that case neither could have caused the other, so nothing is at stake. Causality survives intact. It is only the idea of a universal "now" spread across space that dies.
Time and distance are tied together by the one speed everyone agrees on, so if time is not universal, distance cannot be either. Here is the sum, and it needs nothing we do not already have.
A rod of length L0 lies at rest in your frame, and something flies past it at speed v. You time the flight from nose-passes-one-end to nose-passes-the-other: t = L0/v. The flying observer times the same two events, but they are present at both of them with a single clock, so theirs is the clock that runs slow: they record t0 = t/γ. They work out the rod's length the only way anyone can — speed multiplied by the time it took to go past: L = v × t0 = v × t/γ = L0/γ. Same speed, less time recorded, shorter rod. So an object moving past you is measured to be shorter along its direction of motion, by exactly the same factor γ:
The best evidence for all of this falls on your head continuously, and it is worth doing the arithmetic yourself.
High in the atmosphere, about 15 km up, cosmic rays from space smash into air molecules and create particles called muons. A muon is a heavy, unstable cousin of the electron. Left alone, half of any batch of muons decays in 1.523 microseconds. They are created moving downward at about 0.998c.
Ten thousand muons are created 15 km up, travelling downward at 0.998c — call it 3 × 108 m/s. Half of any batch decays every 1.523 microseconds.
Work out the trip time, and how many half-lives that is. Before doing any relativity: how many of the ten thousand should reach sea level?
Zero — and that is the prediction the classical calculation makes, unambiguously. Now hold a detector at sea level. Muons pour through it, roughly one per square centimetre per minute, day and night, right now, through this page and through you. A prediction of none against an observation of steady rain is not a small error; it does not even have a ratio. Here is the sum done both ways.
Run it with relativity switched off and you will get zero survivors out of ten thousand, every time. Switch relativity on at 0.998c and thousands come through — but wind the speed back to 0.9c and the simulation still shows you none: γ is only about 2.3 there, and the muons need γ near 16 to survive the fall. Relativity does not rescue slow muons, and the sky does not send us any. The muons hitting your roof are not a subtle confirmation of a subtle theory; they are a factor of a billion.
Nothing you own moves fast enough for length contraction to matter, or fast enough for time dilation to be noticeable. But you can find out exactly how close to c you have to get before anything happens. Work out γ for a passenger jet at 900 km/h (0.00000083c): γ = 1.00000000000034. Over an eight-hour flight, the speed effect alone makes the passenger age about ten nanoseconds less than someone on the ground — though a real passenger is also ten kilometres higher, which pushes the other way by about thirty nanoseconds, so they actually land fractionally older. Then work out γ for 0.5c — 1.155, a 15% effect — and for 0.9c, 0.99c, 0.999c. Plot γ against v/c on graph paper. The curve is flat and boring until about 0.5c and then goes vertical. That shape is why three centuries of superb physicists never suspected any of this.
| Idea | What it says |
|---|---|
| Postulate 1 | The laws of physics are identical in every non-accelerating frame. No experiment reveals absolute motion. |
| Postulate 2 | Light travels at c for everyone, whatever their motion or the source's. Forced on us by Maxwell and confirmed by Michelson–Morley. |
| γ | γ = 1/√(1−v²/c²). Derived from the light clock with Pythagoras alone. Equals 1 to fifteen decimals at everyday speeds. |
| Time dilation | A moving clock is measured to run slow by γ. Symmetric: each of you sees the other's clock as the slow one, and both are right. |
| Simultaneity | "At the same time, over there" is not a property of events; it depends on the observer. This is what resolves the paradoxes. Causal order is still absolute. |
| Length contraction | Moving objects are shorter along their motion by γ. The muons reaching sea level are the proof, and the margin is a factor of a billion. |