A body gives off light.
What does it lose?

The 1904 shelf, and the one thing that is not on it

Eight of the nine results below were available to a careful reader at the end of 1904. The ninth is the June 1905 transformation used at step 3, and it is marked as an import because it is one.

The dates on these cards come from standard bibliographies. No one here has checked them against the volumes, and the shelf marks each card as awaiting verification.

The nagging fact

A body that gives off light in one direction recoils. If it gives off equal pulses in opposite directions it does not recoil, and yet the two pulses carry different energies as a moving observer measures them.

The first honest question

If you already know how the energy of a light pulse changes between frames, what does the conservation of energy force you to say about the body that gave it off?

01 / Start with a body that does nothing

Where can the energy have gone?

Put a body at rest in front of you and let it give off light: two equal flashes, one to the left and one to the right, at the same moment. Because the flashes are equal and opposite, whatever push one gives the body the other takes away. The body does not recoil. It sits exactly where it sat, at rest, before and after.

Energy is conserved, so the energy carried off by the flashes came out of the body. But the body has not slowed down, because it was not moving, and it has not moved. Nothing you can see about it has changed. A quantity has left and no visible property has altered to account for it.

02 / Make a prediction

Is the body lighter, or is that a category error?

Before going on, commit to an answer. After the flashes have left, is the body’s mass the same as before, smaller, or is the question malformed because mass is not the sort of thing energy can be taken out of? There is a respectable case for each in 1904, and the argument below only decides between them under premises you will be able to see.

03 / Describe the same event twice

Two accounts of one emission

Describe the same emission a second time, from a frame gliding steadily past at speed v. Nothing about the body changes; you have changed only where you are standing. Write down the body’s energy before and after in each account, four quantities in all.

You cannot evaluate any of the four, because each contains the body’s absolute energy content, which no one knows, in 1904 or now. That is why the second account earns its place: the quantity you cannot supply appears in both accounts, so a subtraction removes it.

The two accounts do not agree about the light, though. A given pair of flashes is measured to carry different total energy depending on the frame it is measured in, and by how much is fixed by the speed and the direction. That is the imported step. It comes from section 8 of the relativity paper, received in June of the same year, and it is the one thing in this route that a reader standing at the end of 1904 could not reach for. It is on the shelf below, marked as an import, with where it came from.

04 / Subtract

What survives when the unknowns cancel

Take the difference between the two accounts before the emission, take it again after, and subtract one from the other. Every absolute body energy disappears, because each appears once in each account. What is left on one side is a difference of differences, and on the other side the light energy multiplied by a factor that depends only on the speed.

The quantity that survives is the difference between what the moving observer and the resting observer say the body’s energy is, and how that difference changed when the light left. For a body whose speed never changed, that is its energy of motion, plus whatever fixed offset separates the two accounts.

05 / Read the coefficient

A number where a category error predicted none

Let the second observer glide past slowly and the speed factor simplifies. The paper prints the result on its last page, in the notation it was set in, where L is the energy given off and V is the speed of light:

Now put that beside the energy of motion you already had on the shelf, one half of the mass times the speed squared. The two expressions have the same shape, and in the place where a mass belongs stands the energy given off divided by the square of the speed of light. The body behaves, for every purpose that mass is measured by, as though it lost that much of it.

First write the result as the paper prints it, in its own letters.

Now put a number to it. The coefficient is small in the units of everyday things, and working one case shows how small.

06 / Check it against the world

Where the paper says to look

What the result says depends on what you weigh. In the ledger below one joule of light leaves a body: draw the boundary around the body alone, around the light, or around both, and read the mass that goes with it. Its cards put the same rule to radium, the Sun, coal, a candle and a year of a light bulb.

// evaluateBoundaryLedger
const bodyDeltaM = -emittedEnergy / (c * c);
const sysDeltaM = disposition === "retained" ? inputEnergy / (c * c) : 0;

// evaluateFourMomentum
const mSquared = (totalEnergy / c)^2 - p^2;
const invariantMass = Math.sqrt(Math.max(0, mSquared));

The paper says only that a test is not ruled out for bodies whose energy content changes a great deal, and names radium salts. The cards above are later tests of another kind: in 1932 Cockcroft and Walton set the energy released when lithium nuclei break apart beside the mass lost, and in 1933 Bainbridge weighed those nuclei with a mass spectrograph. They test the result, and they were on no one’s shelf when the paper was written.

07 / Try it yourself

Four pieces of the argument to work by hand

First the step that makes the direction of the light irrelevant. Any correct form is accepted: the checker compares your expression with the answer at sample values of L, v and V, not by matching text.

Next the 1906 route, the side door below. The number is small enough that the sign is the only thing you could see.

Then the step that decides where the mass is read: at a finite speed the quotient is not yet the mass.

Last, the choice the whole argument rests on, in your own words.

Where this enters the paper

The paper is three pages and its title is a question rather than a claim. It sets out the two accounts, performs the subtraction, takes the slow-speed limit, and states the conclusion in a single sentence. Its German text is on this site, not yet reviewed; the English translation is not written yet.