One current, two stories.
Which one is the moving one?

The nagging fact

Push a magnet into a coil of wire, or move the coil over the magnet at the same relative speed, and the needle deflects by the same amount. The accepted theory of 1904 accounts for the two cases by two different mechanisms, and no measurement tells them apart.

The first honest question

If every measurement you can make comes out the same in both cases, what are you still entitled to say about which one is moving?

01 / Start where the paper starts

A magnet, a coil, and a needle that moves

Hold a coil of wire still and push a magnet into it: the needle deflects. Now hold the magnet still and move the coil over it instead, at the same relative speed: the needle deflects by the same amount. Every measurement you can make on the two arrangements agrees.

The accepted theory of 1904 does not describe them the same way. In the first case the moving magnet is said to produce an electric field in the space around it, and that field drives the current. In the second the field around the magnet is said to be unchanging, and the current arises instead because the wire is moving through it. Two mechanisms, two descriptions, one needle that cannot tell them apart.

02 / Decide whether that bothers you

An asymmetry in the telling, not in the measuring

This is not a crisis. Nothing is predicted wrongly. The two accounts agree on every number, and a physicist who shrugs and uses whichever is convenient will never be caught out by an experiment.

What is odd is that the theory distinguishes two situations that no measurement distinguishes. To tell the stories apart you have to say which object is really moving, and the only thing that could settle that is motion relative to the medium light travels in. Attempts to detect that motion had been made, with increasing care, and had found nothing.

03 / Make a prediction

Can you keep both of these at once?

Here are two statements. First: the laws of physics look the same to anyone moving steadily, so there is no experiment that reveals who is really at rest. Second: light travels through empty space at a definite speed that does not depend on how fast its source was moving when it left.

Before reading on, decide whether they are compatible. If I am gliding past you at half the speed of light and we both watch the same flash, the ordinary way of adding speeds says we must get different answers for how fast it travels. Either one of the two statements is wrong, or something in the ordinary way of adding speeds is.

04 / Find the assumption

What does it take to say two distant things happened at once?

Stop and ask what the claim even means. Two events at the same place happening together is something you can see. Two events far apart happening together is not: you have to compare two clocks, and to compare them you have to have set them, and to set them you have to send something between them that takes time to arrive.

So “at the same time” for distant events is not something you observe. It is something you establish, by a procedure you choose. Write the procedure down and the consequence follows at once: two observers moving relative to each other, each setting their clocks by the same rule, do not end up agreeing about which distant events are simultaneous. Neither has made a mistake.

05 / Build the map

What the definition forces

Now the work is mechanical, and this is the step that the story usually skips. Take the two statements from step 3 as given, take the clock-setting procedure from step 4, and ask what relation must hold between one observer’s coordinates and the other’s. Add that space and time are the same everywhere and in every direction, and the answer is forced: only one set of relations keeps both principles and reduces to the familiar one at everyday speeds.

Work it through rather than read it. The result is often presented as a formula handed down; here it is built from a measurement procedure you can carry out, and nothing enters it beyond the two principles, the clock-setting rule, and that sameness of space and time.

06 / Read off the consequences

Moving clocks, moving rods, and added speeds

A clock moving past you reads less elapsed time than yours between the same two events. A rod moving past you measures shorter along its direction of travel. And speeds no longer add the way they used to, which is the answer to the puzzle in step 3: the flash comes out at the same speed for both of us. Neither measurement is wrong; we disagree about the times and distances that go into it.

First the clock, as a formula.

Then the rod, on numbers.

07 / Meet the serious rival

Lorentz gets the same formulas, and is not refuted

Keep the light medium. Say that a body moving through it genuinely contracts, and that a clock moving through it genuinely runs slow, and that what a moving observer calls time is a bookkeeping device rather than the real thing. Work it through and you reach the same equations, and every experiment above comes out the same way.

This route does not tell you that account is wrong, and it would be dishonest to. Within the scope of these measurements the two are equivalent, and no experiment listed here separates them. What differs is the bookkeeping: one account posits a medium nobody can detect and then adds contraction and local time as properties of motion through it; the other takes the two principles of step 3 and gets the same relations with nothing else assumed.

08 / Check it against the world

What had already been measured

Light in moving water is dragged along, but only partly, by a fraction measured in 1851 and awkward for the theories of its day. The new way of adding speeds gives that fraction with nothing added. A star’s apparent position shifts through the year by an amount known since 1729, and the same transformation gives the shift and the accompanying change of frequency together, from one relation rather than two.

Those were measured before the paper, and the new kinematics gives them. Section 4 adds a consequence nobody had measured: a clock that moves runs slow, by a factor the paper writes down. The laboratory below computes it; the measurement came thirty-three years later.

/**
 * Clock loss per coordinate second: 1 - 1/gamma = beta^2 / (1 + sqrt(1 - beta^2)).
 * printedSecondOrder is (1/2) beta^2.
 */
export function dilationLossPerSecond(
  beta: number,
): KinematicResult<Readonly<{ exact: number; printedSecondOrder: number }>> {
  const bad = refuseBeta(beta);
  if (bad) return bad;
  if (isMode1904()) refuseModernSurface("gamma");
  const root = Math.sqrt(1 - beta * beta);
  return ok({
    exact: (beta * beta) / (1 + root),
    printedSecondOrder: 0.5 * beta * beta,
  });
}

/** Each inertial frame reports the same dilation factor for the other clock's rate -- the
 * symmetric statement this bead requires, without constructing a desynchronized separated-
 * reading comparison (that needs events.ts's simultaneity apparatus, out of this bead's Scope
 * to build). */
export function reciprocalRates(beta: number): KinematicResult<ReciprocalRatesSummary> {
  const g = gamma(beta);
  if (g.status !== "value") return g;
  return ok({ dilationFactor: g.value });
}

09 / Try it yourself

Four pieces of the argument to work by hand

The first three put the paper’s relations to work, in the modern c where the paper writes V. The checker evaluates what you write at many values and compares numbers, not text, so any correct way of writing an answer is accepted.

First, how two speeds combine, as section 5 finds.

Next, a clock’s speed from what it reads and what the resting clocks read.

Then the light clock, a later picture of the same factor.

One to put in your own words, from step 04.

Last, a prediction to make before you change anything, in the laboratory at step 08.

The 1904 shelf

Nothing here is imported from later. The most interesting card is Lorentz’s, which is on the shelf as a live alternative rather than as a foil.

Where this enters the paper

The paper is thirty-one pages in two parts. The magnet and the coil are its opening paragraph, the clock-setting procedure is section 1, simultaneity is section 2, the construction is section 3, clocks and rods are section 4, velocity addition is section 5, and the electrodynamic half from section 6 onwards carries the field transformations, Doppler and aberration, and the dynamics of the electron. Its German text and an English translation are on this site.

Two doors lead to the same place, the map of section 3: the paper’s own construction from the clock-setting rule, and the map built one requirement at a time in a laboratory, a programmer’s way in. Both arrive at the same equation.

Rebuilding this paper is a different exercise from reading it, and long enough to be worth doing on its own. The capstone carries both halves, the kinematics and the electrodynamic sections that follow it, and marks the two steps the paper chooses rather than derives: Rebuild the electrodynamics of moving bodies. It is optional, and nothing above depends on it.