Physics · Module 20
The last module gave up absolute time and space to save the two postulates. This one collects the bill: energy and mass turn out to be the same thing, and — the promise kept — magnetism turns out to be nothing but electricity, seen from a moving frame.
Take a proton and push it with a steady force F. Newton (Module 3) says the acceleration is F/m, where F is the force and m the mass, constant, so the speed rises without limit: keep pushing and you will pass c, the speed of light, in due course.
You will not. But it is worth being careful about why not, because the usual one-line explanation — 'mass increases as you go faster' — is misleading. The proton's mass does not change. What changes is how much speed you get for each extra unit of momentum.
What is really conserved and really additive is not velocity but momentum — you built the ledger for it back in Module 2. In relativity the rule changes. Unlike every other result in this module, we take it on trust: the collision argument that forces it is in Feynman's Chapter 16, listed under At source. Momentum is:
A steady force adds momentum at a steady rate — that is what force means, and it stays true. But as v creeps towards c, γ grows without bound, so the momentum can grow forever while the speed barely moves. You are not being resisted by anything; you are simply pouring your effort into a quantity that no longer translates into speed.
And where does the energy go? Into this:
At speeds far below c, γ is almost exactly 1 + ½v²/c², so γmc² becomes mc² + ½mv² + (tiny corrections). The second term is the kinetic energy you have used since Module 5. The first is new, enormous, and always there. A one-kilogram brick sitting on a table contains 9 × 1016 joules — roughly the energy released by a large power station running for three years.
This is not a licence to extract it. Getting energy out of mass requires a process that actually converts one into the other. Only nuclear processes do it fast enough to matter, and those are Module 24. But the accounting is real, and it is running everywhere: the Sun converts about four million tonnes of its mass into sunlight every second, and it is doing so as you read this.
Watch the two speed traces separate. They agree perfectly at the start. Newton is not wrong; he is a low-speed approximation. Then the relativistic trace bends over and creeps along just under c while Newton's shoots past it and away. Meanwhile the kinetic energy meter climbs without limit. At the Large Hadron Collider protons reach 0.999999991c; getting from there to 0.9999999999c would take vastly more energy again, and c itself would take infinite energy. It is not a wall. It is an asymptote — a line the curve creeps closer and closer to but never reaches.
Module 15 ended by claiming that the magnetic force is not a separate force of nature at all — that it is the electric force, seen from a moving point of view. Now we can see why. The shape of the argument is clear, and most of it is arithmetic you can follow step by step; the three borrowed results are flagged when they arrive.
Here is the set-up. A straight copper wire. Inside it, a lattice of positive metal ions — a rigid grid locked in place — with some number of positive charges per metre. Threading through them, a river of electrons drifting along at speed u, with an equal number of negative charges per metre. The wire is electrically neutral — the two densities cancel exactly, which is why a current-carrying wire does not attract bits of paper.
Outside the wire, at some distance, a test charge q, positive, moving parallel to the wire at speed v in the same direction as the electron drift.
In the laboratory frame, the story is Module 15's. The wire is neutral, so there is no electric field, so no electric force. But there is a current, so there is a magnetic field wrapping round the wire, and the moving charge q feels F = qvB. Work out which way it points. The electrons drift one way, so the conventional current runs the other way; and a positive charge moving with the electrons is, in effect, a current running against the wire's current. Opposed currents repel. So q is pushed away from the wire.
Now ride along with the charge. In this frame the test charge is at rest. A charge at rest feels no magnetic force at all — F = qvB with v = 0 is zero, no matter how strong B is.
But the charge is certainly still being pushed away from the wire. Whether it drifts away is not a matter of opinion: if the two frames disagreed about which way it moved, relativity would be dead on the spot. So in this frame there must be a force, and the only kind left is electric — which requires the wire to be charged. And a moment ago it was neutral.
The wire is exactly neutral in the laboratory. Now you view it from a frame moving parallel to it. The positive ions and the negative electrons are moving at different speeds in this new frame, so their spacings contract by different amounts.
What happens to the wire's neutrality?
Charge is conserved — every electron is still an electron and still carries the same charge; that much is absolute. What is not absolute is how many of them fit in a metre, because the metre is negotiable.
Look at what the last ledger box is telling you. Two calculations produce forces in a fixed ratio of exactly γv, for every setting of both sliders — precisely the factor by which a transverse force transforms between the two frames. Divide the electric one by γv and you have qvB, to every decimal place the bench shows. They share no equations at all. One is built entirely out of Coulomb's law and length contraction. The other is built entirely out of the magnetic field of a current. The magnetic force is not like a relativistic electric force. It is one.
And now go back and reread Module 15's rule about which way the force points, and Module 17's two stories for the rod on the rails. They were never two things. Whether you call an effect "electric" or "magnetic" depends on how fast you happen to be going. Change frames and some of one turns into some of the other, and the total is what stays fixed.
Drift speeds in a real wire are absurdly small — a fraction of a millimetre per second, as we worked out in Module 14. Take u = 0.05 mm/s and a test charge drifting past at v = 1 m/s, and the factor uv/c² is around 10−21. How can an effect that tiny be the whole of magnetism?
Because of how many charges there are. A centimetre of copper wire contains something like 1022 conduction electrons. An imbalance of one part in 1021, applied to 1022 charges, leaves you with real, countable, uncancelled charge — and the electric force is so overwhelmingly strong (Module 13 measured it as 1039 times gravity) that a leftover that small is still enough to run every motor on Earth. Magnetism is weak because it is a relativistic correction; it is noticeable at all only because electricity is so absurdly strong to begin with. The bench cranks u up to a substantial fraction of c purely so that you can see the spacings change. The physics is identical; only the visibility is faked.
It is fair to want an application, so here is the one that would fail without any of this.
A GPS satellite orbits about 20 000 km up at roughly 14 000 km/h, carrying an atomic clock. Your phone works out where it is by comparing the arrival times of signals from several satellites. Because the signals travel at c, timing errors are distance errors: get the time wrong by one microsecond and you get the position wrong by 300 metres.
Two relativistic effects act on those clocks, and they pull in opposite directions.
| Effect | Why | Size, per day |
|---|---|---|
| Special relativity — time dilation | The satellite is moving fast relative to you, so its clock runs slow. This module. | −7 microseconds |
| General relativity — gravitational time dilation | The satellite is higher up in Earth's gravitational field, where clocks run fast. Einstein's 1915 theory, which we have not done. | +45 microseconds |
| Net | The second wins. | +38 microseconds |
Thirty-eight microseconds a day sounds negligible. It is not: at c that is about 11 kilometres of position error, accumulating every single day — roughly 8 metres every minute. Uncorrected, the error would be wider than a road within a minute, longer than a street within ten, and around 2 kilometres off by lunchtime. The satellites' clocks are therefore deliberately built to tick at a slightly wrong rate on the ground, so that they tick correctly once in orbit.
Every time your phone finds itself, it is applying Einstein's corrections. Both of them.
That second row is general relativity, and it is a different theory from the one in this module. Special relativity, which is all we have done, handles observers moving at constant velocity. Einstein spent the next ten years extending it to acceleration and gravity, and arrived at something far stranger. Gravity is not a force at all. Mass bends spacetime, and a falling object is simply going as straight as it can through a bent geometry. That theory reproduces every result of Module 6's gravitation, and then predicts things Newton cannot: the precise orbit of Mercury, the bending of starlight past the Sun, black holes, and the gravitational waves finally detected in 2015 after a century of waiting. That is a course of its own, and worth taking.
| Idea | What it says |
|---|---|
| Mass–energy | p = γmv, E = γmc². Energy is finite at zero speed and infinite at c. Mass is a form of energy. Module 24 spends it. |
| Magnetism | Change frames and a neutral current-carrying wire acquires charge, because the two rows of charges contract by different factors. The electric force in the charge's frame, divided by γv to bring it back to the lab, is exactly qvB. Magnetism is electricity in motion. |