The last of the 1905 papers is the famous one, and it is barely a paper. It runs about three pages, contains no new physics, and follows entirely from the relativity paper Einstein had submitted three months earlier. He seems to have noticed the result afterward, worked it out, and sent it in as a short note. The equation everyone knows does not appear in it in the form everyone knows.

The Problem

The relativity paper had reorganized space and time, but it left an obvious loose end. If velocities do not add the way Newton said, kinetic energy cannot behave the way Newton said either. Momentum, energy, and mass are all defined in terms of motion, and motion had just been redefined.

The specific question Einstein asks is narrow. Take a body at rest that emits light in two opposite directions, so that it stays put — no recoil, no change in velocity. It has clearly lost energy. Does it lose anything else?

Background Science

Two ideas were already in circulation, which is worth being clear about, because Einstein's result did not come from nowhere.

Electromagnetic radiation carries momentum. This follows from Maxwell and had been established for decades — light exerts pressure on what it hits. And several people had noticed that a charged object should therefore appear to resist acceleration more than its material would suggest, since its own field carries momentum too. Hasenöhrl had published a relation between the energy of radiation in a cavity and an effective mass, off by a numerical factor. Poincaré had discussed a momentum density for the electromagnetic field.

All of this treated the effect as electromagnetic — a property of charged bodies and fields, not of matter in general. That is the assumption Einstein removes.

What He Did

He runs the light-emission scenario twice, once in the rest frame of the body and once in a frame moving relative to it, and compares the bookkeeping.

In the rest frame, the body emits total energy E and remains at rest. In the moving frame, the emitted light is Doppler-shifted — one beam blueshifted, one redshifted — and the shifts do not cancel in the energy accounting the way they cancel in the momentum accounting. The energy carried away, measured in this frame, exceeds E by a factor that depends on the frame's velocity.

The body's velocity is unchanged, so if energy is conserved, the discrepancy has to come from somewhere. The only remaining term is the body's kinetic energy, and the only way kinetic energy can change while velocity does not is if the mass has changed. Working it through:

Δm = E/c²

Einstein writes it as a statement about how much mass a body loses when it radiates, not as a general identity. E = mc² is a rearrangement someone else would popularize later.

Why It Mattered

The claim is that mass is not a fixed property of an object. Heat something and it gets heavier. Compress a spring and it gets heavier. The increase is absurdly small — c² is a large number to divide by — which is why nobody had noticed.

It also explains something that had been a genuine puzzle. Radioactive materials emit energy continuously and appeared to violate conservation, which had led to speculation about the ether supplying it. The energy was coming from mass, in quantities too small to weigh.

Einstein closes by suggesting radium salts as a possible test, since their energy output is large enough that the corresponding mass change might be detectable. The first real confirmation came in 1932, from Cockcroft and Walton splitting lithium.

Some Thoughts

What I find most interesting is how tentative the ending is. The final sentence proposes that the theory might be testable if a substance can be found whose energy content varies enough to measure. He is not announcing a law of nature; he is asking whether anyone can check.

The title is also a question — whether a body's inertia depends on its energy content — and the paper never quite converts it into a declarative. The derivation applies to a specific case, radiation emitted from a body at rest, and Einstein extends it to "all energy" in a sentence, without a general proof. That generalization is the actual content of the famous equation and it arrives as an aside.

There is a broader pattern across these four papers that this one makes clearest. In every case Einstein's contribution is a consequence that other people had the ingredients for and did not draw. Planck had quantized energy but kept it in the emitters. Lorentz had the transformations but kept the ether. Hasenöhrl had radiation contributing to inertia but kept it electromagnetic. The move each time is not to discover a new ingredient but to remove a qualification that everyone had assumed was load-bearing.

I do not think that is a method you can teach, but it is at least a recognizable habit, and it is the thing that makes 1905 look like one project instead of four.