One more consequence of the paper on electrodynamics has occurred to me. The principle of relativity, together with Maxwell’s equations, requires that mass be a direct measure of the energy contained in a body. The idea is amusing and enticing; but for all I know, the Lord may be laughing over the whole thing and leading me around by the nose.

— Albert Einstein, letter to Conrad Habicht (summer 1905)

In September 1905, three months after the relativity paper, Einstein sent the Annalen der Physik a supplement of just three pages, with a title in the form of a question: “Does the inertia of a body depend upon its energy content?”. The answer was yes, and the conclusion, written in words and with other letters, is the most famous formula in physics:

$$\boxed{E=mc^2.}$$

The argument, simplified

Einstein imagined a body at rest that emits, at the same moment, two equal pulses of light in opposite directions, each with energy $\frac E2$. Since the pulses balance each other, the body stays at rest. So far, nothing strange: the body has lost energy $E$.

Now the same scene is viewed by someone passing at a small speed $v$. For this observer the body is moving, before and after, always with the same speed $v$. But by relativity the light emitted forwards looks slightly more energetic and the light emitted backwards slightly less, by a factor that, for small $v$, gives in total

$$E\left(1+\frac{v^2}{2c^2}\right).$$

Total energy must be conserved for this observer too. Besides $E$, the body has lost an extra amount $E\frac{v^2}{2c^2}$, and that loss can only have come from its energy of motion, $\frac12mv^2$. Since the speed did not change, what changed was the mass:

$$\frac12(\Delta m)\,v^2=E\,\frac{v^2}{2c^2}\quad\Longrightarrow\quad \Delta m=\frac{E}{c^2}.$$

A body that loses energy $E$, in any form, loses mass $\frac E{c^2}$. Einstein concluded: “the mass of a body is a measure of its energy content”. Mass is stored energy.

How much is that?

The number $c^2$ is enormous: $(3\times10^8)^2=9\times10^{16}$ m²/s². That is why a small mass corresponds to a gigantic energy. A single gram:

$$E=0.001\ \text{kg}\cdot9\times10^{16}\ \text{m}^2/\text{s}^2\approx9\times10^{13}\ \text{J}.$$

That is the energy of about 21,000 tonnes of TNT, more than the Hiroshima bomb, or all the electricity a town of about 10,000 people uses in a year.

Conversely, an everyday energy corresponds to a ridiculously small mass. When you heat a litre of water from 20 °C to 100 °C, the water gains about 335,000 J, and its mass increases by

$$\Delta m=\frac{335\,000}{9\times10^{16}}\approx3.7\times10^{-12}\ \text{kg},$$

some 4 billionths of a gram. No scale notices. That is why, for centuries, mass seemed to be exactly conserved.

The Sun and the stars

Where mass really turns into energy is in the nucleus of the atom. At the centre of the Sun, four hydrogen nuclei fuse into one helium nucleus. But the books do not balance:

$$4\times1.00783\ \text{u}=4.03130\ \text{u}\quad\text{against}\quad 4.00260\ \text{u}\ \text{for helium}.$$

$0.0287$ u is missing, about 0.7% of the mass. That missing mass went out as energy, $\Delta m\,c^2$, and that is why the Sun shines. The Sun emits $3.8\times10^{26}$ watts; dividing by $c^2$, it loses

$$\frac{3.8\times10^{26}}{9\times10^{16}}\approx4.3\times10^{9}\ \text{kg per second},$$

more than 4 million tonnes a second. Even at that rate, in 4.6 billion years it has lost less than a thousandth of its mass.

Energy and motion

The full relativistic formula for a body of mass $m$ moving with speed $v$ is

$$E=\gamma\,mc^2,\qquad\gamma=\frac1{\sqrt{1-v^2/c^2}}.$$

At rest, $\gamma=1$ and what remains is $E=mc^2$, the rest energy. For small speeds, $\gamma\approx1+\frac{v^2}{2c^2}$, and

$$E\approx mc^2+\frac12mv^2.$$

The second term is Newton’s old kinetic energy. Classical physics is a special case of Einstein’s, for slow things. And since $\gamma$ grows without limit as $v\to c$, it would take infinite energy to bring a body with mass up to the speed of light: nothing with mass reaches it.

Exercise: a 1,000 kg car at 108 km/h has how many joules of kinetic energy? How much does that increase its mass?

With $v=30$ m/s, $K=\frac12\cdot1{,}000\cdot30^2=450{,}000$ J. The mass increase is $\frac{450\,000}{9\times10^{16}}=5\times10^{-12}$ kg, five billionths of a gram. At everyday speeds, mass practically does not change.

Exercise: what is the rest energy of an electron, of mass 9.11 × 10⁻³¹ kg, in electronvolts?

$E=9.11\times10^{-31}\cdot(3\times10^8)^2\approx8.2\times10^{-14}$ J. Dividing by $1.602\times10^{-19}$ J/eV gives about $511{,}000$ eV, or 0.511 MeV. When an electron meets its antiparticle, the positron, both vanish and two gamma rays of exactly 0.511 MeV each are left. That is how PET scans in hospitals work.

Sources: Albert Einstein, “Does the inertia of a body depend upon its energy content?”, Annalen der Physik 18 (1905), in Einstein’s Miraculous Year, ed. John Stachel (Princeton, 1998) · David Bodanis, E=mc²: A Biography of the World’s Most Famous Equation (Walker, 2000).

All topics on Albert Einstein · ← Special relativity: a clock made of light · General relativity: from the lift to the Sobral eclipse →

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