E = mc²
The world's most famous formula is not a recipe for manufacturing energy out of nothing. It's an exchange rate between two quantities once thought unrelated, mass and energy, with c² as the conversion factor. Understanding this formula means understanding why it explains neither the bomb nor motion, and why the line everyone recites isn't even the complete equation.
0The pieces of the formula
Three symbols, one accounting system. Each answers a precise question, and the last, c², is the one most misunderstood.
| Piece | Role |
|---|---|
| E | The rest energy of the body, expressed in joules (J). |
| m | The rest mass of the body, expressed in kilograms (kg). |
| c | The speed of light in vacuum, 299,792,458 m/s: an exact value since 1983, since the metre is itself defined from it. |
| c² | The conversion factor between the two units, about 8.988 × 10¹⁶ m²/s². |
1What Einstein actually wrote
The formula everyone recites is not the sentence Einstein published first. His original version fits in three pages, and it runs the other way.
Einstein's fourth paper of 1905, "Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?" (Does the inertia of a body depend on its energy content?), appears in the Annalen der Physik. Three pages, one conclusion: if a body emits an energy L in the form of radiation, its mass decreases by L/c². The modern notation E = mc² only settles in during later work.
Einstein doesn't start from a mass that would need converting: he starts from a body that radiates light, and shows that this radiation costs it mass. It's the reverse of the popular bomb narrative, which imagines mass being "converted" all at once. Einstein ends his paper by suggesting that radium salts, known to emit energy spontaneously, might allow the idea to be tested.
2The thought experiment
Einstein's argument assumes nothing more than special relativity, published three months earlier. It fits in one page.
A body at rest emits two identical light pulses in opposite directions: its motion doesn't change, the symmetry of the two emissions cancels out exactly. Seen from a frame moving relative to the body, however, the Doppler effect makes the two pulses unequal, one is blueshifted, the other redshifted.
Writing the energy balance in this second frame, a term appears that has exactly the form of a kinetic energy, that of a body whose mass has decreased by L/c². Yet the body has done nothing but emit light: it's this emission, and this alone, that forces the loss of mass. The argument rests on no additional assumption, only on the special relativity Einstein himself had just published.
3The exchange rate
c² is worth about 90,000,000,000,000,000: converting even the smallest mass into joules gives a dizzying number. The lab below does the math for you, with everyday objects.
4The mass defect, or energy comes from somewhere
In a nucleus, the mass of the whole is less than the sum of the masses of the parts. That difference has a name, and it's the only real source of nuclear energy.
Assemble protons and neutrons into a nucleus, and the nucleus weighs less than the sum of its separate constituents. The difference is called the mass defect, and it is exactly equal to the binding energy that holds the nucleus together, converted through c². Matter isn't destroyed: you descend the staircase of binding energy, and the step you descend is paid for in missing mass. The atomic mass unit is worth 931.494 MeV/c².
It's the only mechanism behind every source of nuclear energy, whether fusing light nuclei or fissioning a heavy one: in both cases, the final state weighs a little less than the initial state, and this loss of mass, multiplied by c², is the energy released. The lab below compares this loss, in proportion, to an ordinary chemical reaction.
5The formula nobody recites
E = mc² is only true for a body at rest. It's a special case of a broader relation, almost never quoted outside textbooks.
The complete relation reads E² = (pc)² + (mc²)², where p is the momentum. For a moving body, the total energy reads E = γmc², with the Lorentz factor γ = 1 / square root(1 − v²/c²). Two consequences follow at once. First, the photon: its mass is zero and its energy isn't, E = pc, something the formula E = mc² alone would leave incomprehensible. Second, the rest energy mc² is a floor, not a total: the energy of a moving body is always greater.
When a body is at rest, its momentum p is zero, and the complete relation reduces exactly to E² = (mc²)², hence E = mc². The world's most cited formula is the special case of a more general equation, the case where nothing moves. That's also why it fails to describe the photon, which never stops.
6What the formula doesn't say
E = mc² doesn't explain the atomic bomb. It does the accounting of the energy released, it says neither which nuclei break, nor how.
The mechanism that makes a bomb possible is fission, discovered experimentally by Otto Hahn and Fritz Strassmann in late 1938, then interpreted by Lise Meitner and Otto Frisch in early 1939, followed by the chain reaction. E = mc² lets you calculate how much energy a fission releases, once you know it's happening; it predicts neither the existence of fission nor how to trigger it.
Einstein himself did not work on the Manhattan Project. His role is limited to a letter addressed to President Roosevelt in August 1939, co-signed with Leó Szilárd, warning of the possibility of a weapon of this kind. Attributing the bomb to his formula confuses the accounting tool with the engineering that made it exploitable: neither indictment nor hagiography does justice to that distinction, it's simply factual.
7Where the formula works today
Three places where E = mc² isn't a textbook curiosity, but a working tool used every day, right down to your own body.
Positron emission tomography (the PET scan) exploits annihilation: a positron and an electron meet, annihilate, and produce two photons of 511 keV each, exactly the rest energy of the electron. The hospital machine literally measures mc². Particle accelerators do the reverse: they convert kinetic energy into new particles, into fresh mass.
The most counterintuitive fact remains this one: the mass of the proton comes mostly from the binding energy of its constituents, the quarks, whose own mass supplies only a few percent of the total. In other words, almost all the mass of your body is binding energy, not matter in the naive sense of the word. That's E = mc² read backwards: here, it's energy that manufactures mass.
8The six misconceptions
9Test yourself
Eight questions, one correct answer each time. The explanation appears after your choice.
10The glossary
| Term | Meaning |
|---|---|
| rest energy | the energy a body possesses simply by virtue of its mass, when at rest |
| rest mass | the mass measured in the frame where the body is at rest; it is an invariant |
| relativistic mass | a notion now abandoned by physicists, which made mass "grow" with speed |
| Lorentz factor (γ) | the coefficient that measures the dilation of time and energy as speed approaches c |
| momentum (p) | a quantity combining mass and velocity, conserved in any isolated system |
| frame of reference | the vantage point, moving or not, from which a physical quantity is measured |
| invariant | a quantity that keeps the same value in every frame of reference |
| atomic mass unit (u) | a unit of mass at the scale of the nucleus, defined from carbon-12 |
| electronvolt (eV) | a unit of energy at the atomic scale, the energy gained by an electron crossing one volt |
| mass defect | the difference between the mass of a nucleus and the sum of the masses of its separate constituents |
| binding energy | the energy it would take to split a nucleus into its constituents; it corresponds to the mass defect |
| fission | the splitting of a heavy nucleus into lighter nuclei, releasing energy |
| fusion | the joining of light nuclei into a heavier nucleus, releasing energy |
| proton-proton chain | the sequence of reactions that fuses hydrogen into helium at the core of the Sun |
| annihilation | the meeting of a particle and its antiparticle, which converts all of their mass into radiation |
| positron | the antiparticle of the electron, of equal mass and opposite charge |
| photon | the quantum of light, massless, carrying energy and momentum |
| antimatter | matter made of antiparticles, the mirror of ordinary matter |
| joule (J) | the unit of energy in the International System |
| kiloton of TNT equivalent | a unit of energy used for explosions, one kiloton being worth 4.184 × 10¹² joules |
| accelerator | an instrument that brings particles to high speed, to convert kinetic energy into new matter |
| special relativity | Einstein's theory published in 1905, which unifies space, time, mass and energy for any observer in uniform motion |