Where Does E=mc² Come From? Einstein's Three-Page Paper of 1905

E=mc² comes from a three-page paper Albert Einstein sent to Annalen der Physik in September 1905, titled “Does the Inertia of a Body Depend upon its Energy-Content?”. In it, he imagines a body at rest that sends out two equal flashes of light in opposite directions, and looks at the same event from a moving observer. Energy has to balance for both observers, and the only way it can is if the body’s mass falls by the energy of the light divided by the speed of light squared. The equation in its famous form is not written in the paper at all.

A sentence, not an equation

What Einstein wrote was a sentence. In the 1923 English translation by W. Perrett and G. B. Jeffery:

If a body gives off the energy L in the form of radiation, its mass diminishes by L/c². — A. Einstein, Annalen der Physik 18, 1905

There is no E in it. Einstein used the letter L for energy, and in the German original he did not even write c for the speed of light: he wrote V. The translation modernised the notation. Turn the sentence into symbols, swap L for E, and you get m = E/c², which is E = mc². The famous form came later.

The paper is pages 639 to 641 of volume 18. The journal received it on 27 September 1905 and published it on 21 November 1905. Einstein was 26 and worked at the Swiss Patent Office in Bern.

The letter to Habicht

The paper was a sequel. On 30 June 1905 the same journal had received “On the Electrodynamics of Moving Bodies”, the paper of special relativity. It starts from two ideas: the laws of physics are the same for two observers moving steadily relative to each other, and light travels at the same speed for both.

Some time between 30 June and 22 September, Einstein told his friend Conrad Habicht what had occurred to him next:

A consequence of the study on electrodynamics did cross my mind. Namely, the relativity principle, in association with Maxwell’s fundamental equations, requires that the mass be a direct measure of the energy contained in a body; light carries mass with it. A noticeable reduction of mass would have to take place in the case of radium. The consideration is amusing and seductive; but for all I know, God Almighty might be laughing at the whole matter and might have been leading me around by the nose. — Einstein to Conrad Habicht, 1905 (English translation as quoted by L. B. Okun)

The title of the paper is the question behind that letter. Inertia is how hard a body is to set moving, and that is what mass measures. Does it depend on how much energy the body holds?

The thought experiment with two flashes of light

Take a body sitting at rest. It sends out two equal flashes of light in exactly opposite directions, each carrying energy ½L, so L in all. The two kicks cancel and the body stays where it is. For us, at rest beside it, the body has simply lost energy L.

Now look at the same event from an observer moving past at a steady speed v. Einstein had already worked out, in the relativity paper, that a moving observer measures a different energy for a flash of light. For the two flashes together, the moving observer measures a little more than L: L/√(1 − v²/c²).

By the principle of relativity, energy must be conserved for both observers. Both see the same body before and after, so the difference between their two accounts can only be the body’s energy of motion, its kinetic energy as the moving observer sees it. That kinetic energy drops, by L × (1/√(1 − v²/c²) − 1).

But the body’s speed has not changed. It was at rest before and at rest after. For speeds much smaller than light, neglecting “magnitudes of fourth and higher orders”, as Einstein wrote, the drop is ½ × (L/c²) × v². Kinetic energy is ½mv². Same speed, less kinetic energy: the mass must have fallen, by L/c². The animation in the video walks through this step by step.

Einstein then let go of the light:

The fact that the energy withdrawn from the body becomes energy of radiation evidently makes no difference, so that we are led to the more general conclusion that The mass of a body is a measure of its energy-content. — A. Einstein, 1905

What E=mc² says, and what it doesn’t

Einstein gave the conversion in the units of his day: if the energy changes by L, the mass changes in the same sense by L/9 × 10²⁰, with the energy in ergs and the mass in grams. That large number is c². Today the speed of light is exactly 299,792,458 metres per second.

That is enough to run a 100-watt bulb for about 28,000 years. It works in both directions and for every kind of energy, not only light. Heating a kilogram of water by one degree takes 4,184 joules; the warm water is heavier by less than a ten-billionth of a gram. Far too little to weigh, which is why nobody had noticed.

Before relativity, physics kept two separate laws, conservation of energy and conservation of mass. Writing in 1946, Einstein described the old view: “Heating, melting, vaporization, or combining into chemical compounds would not change the total mass.” Relativity merged the two laws into one.

What the equation does not say matters as much. It is not a recipe: it gives the exchange rate between mass and energy, not a way to release the energy. And in the 1905 paper, m is the mass of a body at rest and E the energy it holds at rest, which is why physicists often write E₀ = mc². Einstein himself, in the same 1946 article: “It is customary to express the equivalence of mass and energy (though somewhat inexactly) by the formula E = mc2.”

How E=mc² was tested

The 1905 paper ends with a hope: “It is not impossible that with bodies whose energy-content is variable to a high degree (e.g. with radium salts) the theory may be successfully put to the test.”

The check came from nuclei. In 1932, at the Cavendish Laboratory in Cambridge, John Cockcroft and Ernest Walton accelerated protons with a few hundred thousand volts and fired them at lithium. Walton saw the bright flashes of alpha particles on a zinc sulphide screen. A lithium nucleus that takes in a proton breaks into two helium nuclei, each carrying 8.6 million electron volts, 17.2 million in all. That energy, Cockcroft said in his Nobel lecture, “could be provided by a diminution of mass of 0.0184 mass units”, and the masses matched within the experimental errors. The 1951 Nobel presentation speech put it plainly: “a verification was provided by this analysis for Einstein’s law concerning the equivalence of mass and energy.”

The most precise direct test so far came in 2005, from Simon Rainville, David Pritchard and colleagues at MIT, NIST and the Institut Laue-Langevin in Grenoble. When a silicon or sulphur nucleus captures a neutron, it gives out a gamma ray. In Grenoble, crystals measured the gamma rays’ wavelengths, which gave the energy E. At MIT, the atoms were weighed before and after, which gave the mass difference. Published in Nature in December 2005, E and mc² agreed to at least 0.00004 per cent, four-tenths of one part in a million: 55 times more accurate than the best previous direct test.

Einstein’s 1905 paper had closed with one more sentence: “If the theory corresponds to the facts, radiation conveys inertia between the emitting and absorbing bodies.” A thought experiment with two imagined flashes of light, confirmed by weighing atoms. And that’s how we found out.

For the prize Einstein did win, and why it was not for relativity, read the story of Einstein’s Nobel Prize.