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§ Phase xi · Mass & Scale · Chapter 02

E = mc²

In the autumn of 1905, in a three-page paper printed on cheap wartime paper in Leipzig, Einstein quietly proposed that mass and energy were the same thing wearing different clothes. The proposal sat almost untouched for thirty-three years. Then a German chemist split a uranium atom, his Austrian collaborator did the bookkeeping in a snowy forest in Sweden, and the world found out what those five symbols had been hiding.

The September 1905 paper was a postscript. Einstein had just finished a longer piece, the one we now call his special theory of relativity, in which a 26-year-old patent clerk had calmly demolished Newton’s notions of absolute space and absolute time. He had done this in three months of evening work after long days examining patents for electric typewriters and clock-synchronization circuits in the Bern post office, sometimes with his infant son Hans Albert in his lap. The longer paper, “On the Electrodynamics of Moving Bodies,” ran to 31 pages and was so unusual in its style that the editor of Annalen der Physik almost asked for revisions before deciding, in a sentence that physicists have been grateful for ever since, that the paper was simply too strange to tamper with.

The little follow-up, which arrived at the editor’s desk three months later, asked a single question. If a body emits energy in the form of light, does its mass go down by exactly the amount of energy emitted divided by the square of the speed of light? Einstein did the calculation in a paragraph. The answer was yes. He titled the paper “Does the Inertia of a Body Depend Upon Its Energy Content?” and signed off with a sentence that read more like a wager than a prediction. “It is not impossible,” he wrote, “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.” Radium was the most exotic substance in any 1905 laboratory. Einstein had no way of seeing the chain reaction, the reactor, or the bomb. He was offering a calculation and a hope.

The equation he wrote down was not exactly E = mc². Einstein actually wrote it as L/V², where L was his symbol for energy and V was his symbol for the speed of light. The shorthand we now use, with E for energy, m for mass, and c for the Latin celeritas, “swiftness”, came later. The content was unchanged. If you remove an amount L of energy from a body, the body’s mass drops by L/V². The conversion factor is enormous because the speed of light, squared, is enormous: nine followed by sixteen zeros in SI units. A single kilogram, fully converted, would yield ninety quadrillion joules, which is roughly the annual electrical consumption of a small country. A single gram would equal the yield of twenty kilotons of TNT. These numbers were, in 1905, completely unreal. No one had ever seen anything close to a one-gram mass loss in a laboratory chemical reaction; chemistry shuffles around at most a few electron-volts per atom, which corresponds to mass changes far below any scale 1905 could measure.

The years rolled on. Einstein became famous, won the Nobel for the photoelectric effect, traveled, gave interviews, fled Germany in 1933. The mass-energy equation became a piece of textbook physics that physicists trusted but could not really demonstrate. Particle accelerators were starting to be built. Cyclotrons in Berkeley were measuring nuclear reactions and finding, dutifully, that the products always weighed a little less than the reactants by exactly the amount predicted by Einstein’s formula. The bookkeeping worked. Energy and mass exchanged at the published rate. Still, no one had seen anything dramatic enough to put the equation in a newspaper.

That changed on the afternoon of December 17, 1938, in a laboratory on the outskirts of Berlin. The German chemist Otto Hahn had been bombarding uranium with slow neutrons for years, trying to make the so-called transuranic elements, atoms heavier than uranium that he and his old colleague Lise Meitner had been hunting since the early 1930s. Meitner had been a Jewish Austrian physicist forced out of Germany that summer; she was now in Sweden, corresponding with Hahn by mail. Hahn’s December chemistry was peculiar. The chemical signatures in his beaker did not match a heavy element. They matched barium, with an atomic number of 56, less than half of uranium’s 92. Hahn wrote to Meitner on December 19 that he had measured the result three times and could not understand it. “Perhaps you can suggest some fantastic explanation,” he wrote. “We understand that it really can’t break up into barium.”

Meitner read Hahn’s letter on Christmas Eve in the small Swedish town of Kungalv, where she was visiting friends. Her nephew, the young physicist Otto Frisch, was there too. The next morning the two of them went for a walk through the snow. Frisch later wrote that Meitner sat down on a fallen log and started doing the arithmetic in the snow on the back of an envelope. She knew the masses. She remembered Bohr’s liquid-drop model of the nucleus, in which a heavy nucleus is held together by short-range nuclear forces but pulled apart by long-range electrical repulsion among its many protons. The two effects nearly cancel, and a uranium nucleus is like a wobbling drop of nearly evaporated liquid. A neutron, absorbed, could push the drop over the edge. It could elongate, neck, and split.

Now we can write down what Meitner saw. A uranium-235 nucleus contains 235 nucleons, 92 protons and 143 neutrons, bound together with an average binding energy of about 7.6 MeV per nucleon. After absorbing a slow neutron and splitting into two medium-sized nuclei, the daughters (typically a barium isotope and a krypton isotope, plus two or three free neutrons) each have a tighter binding energy of about 8.5 MeV per nucleon. The difference, multiplied by the number of nucleons, is about 200 MeV released per fission event. That energy comes from somewhere. It comes from the mass.

Two hundred MeV per atom does not sound like much. Burning a hydrocarbon, by contrast, releases only a few electron-volts per molecule. The ratio is roughly a hundred million to one. That is why a sugar cube of uranium can in principle release as much energy as a freight-train carload of coal. The conversion ratio, in mass terms, is small. Only about a tenth of one percent of a uranium nucleus’s mass is liberated when it splits; the rest stays bound up in the daughters. But that tenth of a percent, multiplied by Avogadro’s number and by c², is the entire energy economy of the nuclear age.

So far, mass to energy. The arrow points one way: a parent nucleus carries internal energy as mass, the products carry it as kinetic energy and heat. But the equation reads in both directions. Energy in the right form can also be turned back into mass, fresh mass, particles that did not exist before the collision. This is the trade that particle accelerators make for a living, and it is what separates a chemist’s laboratory from a physicist’s beamline.

Consider an electron and a positron, the electron’s antimatter twin, hurtling toward each other at nearly the speed of light. Their total energy in the center-of-momentum frame, call it √s, is the budget the collision has to spend. If √s is small, the two particles annihilate into a couple of photons and the energy departs as light. If √s exceeds twice the rest mass of some heavier particle, however, the annihilation can produce that particle and its antimatter twin. At √s near 3 GeV, the collision can produce a J/ψ meson, a bound state of a charm quark and an anticharm antiquark, discovered in dueling experiments in 1974 at Brookhaven and SLAC. At √s near 91 GeV, it can produce a single Z boson, the carrier of the weak neutral current. At √s above 350 GeV, it can produce a top-antitop quark pair, the heaviest known fermion. None of these particles existed before the collision. The kinetic energy of the incoming particles literally becomes their rest mass.

The Large Hadron Collider near Geneva does the same trick with protons. It accelerates two counter-rotating beams of protons around a 27-kilometer ring of superconducting magnets, ramping their kinetic energy until √s for a head-on collision reaches 14 TeV. The protons themselves are not really the players; their internal quarks and gluons are. When two gluons from the colliding protons happen to interact at high enough energy, they can produce, for one fleeting instant, a Higgs boson with a rest mass of 125 GeV, more than a hundred and thirty times the mass of a proton. The Higgs was the design target of the LHC, found at last in July 2012 after fifty years of theoretical prediction. The detectors do not see the Higgs directly; they see its decay products, two photons, four leptons, a clean signature, and they read off the missing energy and momentum. Mass-energy bookkeeping is exact at every step.

E² = (pc)² + (m c²)²

E = √( m²c⁴ + p²c² ) ≈ mc² + p²/(2m) + …

There is one more twist, the one that genuinely surprises physicists when they meet it for the first time. Most of the mass of ordinary matter is not the mass of the particles that ordinary matter is made of. Take a single proton. A proton is a bound state of three valence quarks, two up quarks and one down quark, held together by an arrangement of gluons exchanging color charge inside the proton’s tiny volume. The up quark has a rest mass of about 2.2 MeV/c², the down quark about 4.7 MeV/c². Add the three together and you get about 9.4 MeV/c². The proton, by experiment, has a rest mass of 938 MeV/c². The arithmetic is off by a factor of a hundred. Where do the other 929 MeV come from?

They come from energy. The gluon field inside the proton is a roiling, self-interacting tangle of color charge that carries enormous energy per unit volume. That energy, by the same equation Einstein wrote in 1905, contributes to the rest mass of the bound system. About 99 percent of the mass of any proton you weigh, including every proton in your body, is gluon-field energy, not the bare rest mass of the quarks that sit inside it. The Higgs mechanism, which gives the bare quarks their few-MeV masses, is essentially a footnote to the proton. The proton is mostly a glowing knot of binding energy. You are made of protons. You are, mostly, a slow boil of binding energy in a temporary configuration, and the configuration carries inertia because the energy carries inertia.

We started this chapter with a four-page postscript from a patent clerk in Bern. We end it with a confession that most of what you weigh is not matter but field energy, and that the equation tying the two together has paid for every kilowatt of nuclear power humans have ever extracted and every gram of new particle ever conjured into existence inside a collider. The same single relation describes a uranium nucleus splitting in 1938, the photon coming off when a positron annihilates with an electron in a hospital PET scanner, the Higgs boson appearing for a femtosecond in a CMS detector under the Swiss Alps, and the dark mass of every proton in every star.

Inertia, the property a child first notices when she pushes a wagon and finds that it resists, is just energy in another costume. Einstein saw that in the autumn of 1905 in a small office above a Bern post office and put it on three pages. Meitner saw what it meant in the snow at Kungalv in 1938 and put it on the back of an envelope. The rest of the twentieth century, and the whole of contemporary particle physics, are footnotes to those two short calculations. Next we ask why some particles, the neutrinos, are unaccountably light, and what that smallness tells us about what is hiding above the energy scales we can yet reach.