Episode 24 - E=mc²

Tomlin Hawthorne

Two identical cups of coffee. Same cup, same coffee, same amount, to the last drop. One is piping hot. The other has gone stone cold.

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Put each on a perfect set of scales.

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Which weighs more?

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The hot one.

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Not because of steam, or expansion, or anything like that. Seal them both in airtight flasks if you like. The hot coffee weighs more than the cold coffee, for no other reason than that it's hot.

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The difference is far too small for any scale ever built. But it's there, and it isn't a quirk. It's the plain meaning of the most famous equation in the world. Today we're going to read that equation, slowly, as a sentence. It doesn't say what most people think it says.

Tomlin Hawthorne

Last time you drew yesterday as a world-line, a thread through spacetime, and went out to find the oldest light you could see. If you got as far as the Andromeda galaxy, you caught light that set out before our species existed. And we met the idea that space and time are two directions in one thing, and that all of us travel through it at a single fixed rate.

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Today's episode is about a consequence of that. It took Einstein three pages. It's the one piece of physics that absolutely everybody can quote, and it has more blood on it than any other. So the question is simply: what does E equals m c squared actually say?

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Let me start with how it sounds from outside physics, because artists have been drawn to this equation as to no other.

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In 2005 the composer John Adams wrote an opera called Doctor Atomic. It's set in the desert in New Mexico, in the hours before the first test of the atomic bomb. At the end of the first act, Robert Oppenheimer, who led the project, stands alone in front of the device, and sings a setting of a John Donne sonnet. The one that asks God to batter his heart. Oppenheimer had named the test site Trinity, and said afterwards he thought he'd had that poem in mind.

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The Swiss playwright Dürrenmatt wrote a black comedy called The Physicists, set in a private asylum. One patient believes he's Newton. One believes he's Einstein. The third says King Solomon appears to him. It emerges that none of them is mad. The third is the greatest physicist in the world, who has had himself locked up to keep what he's discovered away from mankind. The other two are spies. It doesn't end well.

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And you'll remember that Brecht wrote his Galileo twice, and changed the ending after August 1945.

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No other equation has had plays written at it. So let's see what it is.

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September 1905. Einstein's relativity paper, the one from episode twenty-one, has been out for three months. He's still at the patent office. And he's noticed something further. He writes to a friend that a consequence has occurred to him, that the argument is amusing and seductive, and that for all he knows the Lord might be laughing at him and leading him up the garden path.

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He sends in a short note. Three pages. Its title is a question: does the inertia of a body depend on its energy content?

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Inertia, you'll recall from Galileo and Newton, is an object's reluctance to change its motion. It's what mass measures. How hard a thing is to shove.

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Einstein's argument is a thought experiment. Take an object, sitting still, and have it give off two flashes of light in opposite directions, equal and opposite, so that it doesn't recoil. It's lost some energy; the light has carried it away. Now look at the same event from a moving train, and insist that the laws of physics work for that observer too. When you do the bookkeeping, you find it only balances if the object's mass went down when it emitted the light.

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It gave off energy, and it got lighter. Not because any substance left it. Light isn't made of substance. The energy itself had weight.

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And the amount is exact. The mass it lost equals the energy it gave off, divided by the speed of light squared. Turn that round and you have it. Energy equals mass times the speed of light squared.

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Now let's read it the way we read Newton's law of gravity, back in episode five. As a sentence.

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E is energy. That's the thing from episode six: nature's ledger, the quantity that's never created or destroyed.

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m is mass. How hard something is to push. How much it weighs.

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And the equals sign is the shocking part. It doesn't say energy can be made from mass. It says they're the same thing. Mass is energy. Mass is what energy looks like when it's sitting still. Before 1905 there were two great conservation laws: matter is never created or destroyed, and energy is never created or destroyed. Einstein says those are one law. There's only one ledger.

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And c squared? The speed of light, times itself. That's not doing anything mysterious. It's an exchange rate. It tells you how many units of energy you get for one unit of mass, the way a rate tells you how many yen you get for a pound.

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And it's an enormous rate. The speed of light is three hundred million metres a second. Squared, it's a nine with sixteen zeros.

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So a tiny amount of mass is a gigantic amount of energy. And, the other way round, an ordinary, everyday amount of energy is an absurdly tiny amount of mass. Which is why nobody had noticed.

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Let me give you the scale in both directions.

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Take one gram of anything. A paperclip. A raisin. If you could convert that gram entirely into energy, you'd get about ninety thousand billion joules. That's roughly the energy of the bomb that destroyed Hiroshima. In fact, in that bomb, the amount of matter that was converted to energy was a little under one gram. A city was erased by the weight of a paperclip, less.

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The Sun shines because of this equation. It's converting mass to light at a rate of more than four million tonnes every second. It's been doing that for four and a half billion years and has barely made a dent in itself.

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Now the other direction, back to the coffee.

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Heat is energy. The hot cup has more energy in it than the cold one: its molecules are jiggling faster. And energy is mass. So the hot cup has more mass. How much more? The extra energy, divided by c squared. Divided by that nine with sixteen zeros. It comes out at less than a billionth of a gram. Hopelessly undetectable. But real.

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And it's the same everywhere you look. Wind up a watch, and the tightened spring has more energy, so the watch is heavier. Charge your phone, and it's heavier. By less than the weight of a bacterium, but heavier. Strike a match: the ash and the smoke and the gases, if you caught every bit of them, would weigh slightly less than the match and the oxygen did, because energy left as heat and light.

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That last one matters. People think this equation is about nuclear physics. It isn't. It's about everything. A burning log is converting mass to energy just as a reactor is. The only difference is the fraction. Burning converts about a ten-billionth of the mass. A nuclear reaction converts about a thousandth. Same equation. Ten million times more of it.

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And here's the one that I think is the real headline, though it wasn't understood until the 1970s.

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Where does your own mass come from? You'd say: from the stuff I'm made of. From atoms. And atoms get their mass from the protons and neutrons in their cores. Fine. And those are made of three smaller things called quarks, which we'll meet in Part Seven.

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But if you add up the masses of the three quarks in a proton, you get about one per cent of the mass of the proton.

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So where's the other ninety-nine per cent?

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It's energy. The quarks are tearing around inside the proton at close to the speed of light, held in by a ferociously strong force. All that energy of motion and of binding, sitting in one place, is what we weigh as the proton's mass. E equals m c squared, read from right to left.

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So when you stand on the bathroom scales, about ninety-nine per cent of the number you see is not the weight of any stuff. It's the weight of energy, trapped in a very small space. You are, overwhelmingly, made of confined motion.

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I find that the most surprising fact in this entire series, and I notice it never makes the tea towel.

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One more thing the equation explains. Why can nothing with mass reach the speed of light?

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Push on a spaceship. It speeds up. That means it has more energy of motion. But energy is mass. So it's now a little harder to push. Push again: more speed, more energy, heavier still. The closer you get to the speed of light, the more of your pushing goes into making the ship more massive and the less goes into making it faster. To reach the speed of light itself would take an infinite amount of energy. The universe has a speed limit, and it's enforced by the ledger.

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Now, the history. And I'd like to tell it through someone who doesn't usually get the first line.

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Lise Meitner was born in Vienna in 1878, into a Jewish family. She was the second woman ever to get a doctorate in physics from the University of Vienna. She studied under Boltzmann. In 1907 she moved to Berlin, and began working with a chemist her own age, Otto Hahn, on the new science of radioactivity.

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The head of the chemistry institute didn't allow women in the building. He said he was worried about their hair catching fire. So Meitner was given a converted carpenter's workshop in the basement, with its own outside door, and told not to come upstairs. To use a lavatory she had to walk to a restaurant down the street. She worked unpaid for the first five years.

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She and Hahn worked together for thirty years. She became a professor, and head of her own physics department. Einstein called her the German Marie Curie.

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In 1938, Austria was absorbed into Nazi Germany, and overnight she lost the protection of her foreign citizenship. That July, with the help of Dutch colleagues, she was smuggled over the border by train. She was fifty-nine. She had two small suitcases and ten marks in her purse. Hahn had given her a diamond ring that had been his mother's, in case she needed to bribe a border guard.

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She ended up in Stockholm, with a desk and no equipment. And she and Hahn kept working together by letter.

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That December, Hahn wrote to her with a result he couldn't understand. He'd been firing neutrons at uranium, the heaviest element. He expected to make something slightly heavier or slightly lighter. And his chemistry was telling him, over and over, that among the products was barium. Barium is about half the weight of uranium. It was as if he'd thrown a pebble at a boulder and the boulder had fallen in two.

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He wrote: perhaps you can suggest some fantastic explanation.

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Meitner was spending Christmas with friends in a village on the Swedish coast. Her nephew, Otto Frisch, also a physicist, also a refugee, came to join her. On the morning after he arrived, she showed him the letter. They went out for a walk in the snow to talk about it. He was on skis. She kept up on foot.

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They sat down on a fallen tree. And they worked it out, on scraps of paper from her pocket.

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Suppose the nucleus of a uranium atom isn't a hard lump. Suppose it's more like a drop of liquid, wobbling. A neutron strikes it, the drop stretches, pinches in at the middle, and tears into two smaller drops. Those two pieces are both positively charged, so they fly apart with tremendous violence.

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Frisch worked out how much energy that would be. It was huge. Where would it come from?

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And Meitner remembered how to work out the masses of nuclei. She did the sum on the log. The two pieces, together, would weigh less than the original uranium nucleus. Less by about a fifth of the mass of a proton.

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A missing bit of mass. So: multiply by c squared.

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It matched. The missing mass, turned into energy by Einstein's equation, was exactly the energy of the flying fragments.

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That was the moment. On a log, in the snow, a woman who'd been driven out of her country and a nephew on skis understood that the atom could be split and that it would release, weight for weight, millions of times the energy of any explosive. Frisch asked a biologist what you call it when a cell divides in two, and borrowed the word. Fission.

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Hahn received the Nobel Prize for the discovery, alone. Meitner's name was left off the key paper, because in Germany in 1939 it couldn't be on it, and afterwards Hahn came to remember the discovery as his.

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She was invited to join the Allied bomb project and refused. She said she would have nothing to do with a bomb. Very few physicists who were asked said no. Element number one hundred and nine is named meitnerium. On her gravestone, in an English country churchyard, her nephew had carved: a physicist who never lost her humanity.

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As for Einstein. In the summer of 1939, a Hungarian physicist, Leo Szilard, drove out to find him at a holiday cottage on Long Island, and explained that a chain reaction in uranium now looked possible. Einstein is said to have answered, in German, that he'd never thought of that. He signed a letter to President Roosevelt, warning that Germany might build such a weapon. It's the letter that started the Manhattan Project. He took no part in the project itself. He said later that signing it was the one great mistake of his life.

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The myth of the week: that E equals m c squared is the formula for the atomic bomb.

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It isn't a recipe for anything. It tells you that mass and energy are the same, and what the exchange rate is. It tells you nothing about how to get the energy out. That needed the discovery of the neutron, of fission, of the chain reaction, and a vast engineering effort. The equation was thirty-three years old before anyone had the faintest idea it could matter outside a laboratory, and Einstein was as surprised as anyone.

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It explains why a bomb is possible. It's about as much to blame as the law of gravity is for a hanging.

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And yet. Somebody still has to decide what to do with what they find out. Which is the question to take away.

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Two years after the war, Oppenheimer gave a lecture in which he said that the physicists had come to know sin. He meant it plainly, he said, and no amount of joking or exaggeration could quite make it go away. And it was a kind of knowledge they could never be rid of.

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What do the people who find things out owe to the people who'll have to live with them? Meitner refused. Oppenheimer led. Dürrenmatt's physicist hid in a madhouse, and the play's bleak point is that it didn't help: what's once been thought can't be unthought. Brecht changed his mind about Galileo.

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I don't think physics can answer this for you. I do think the humanities have had the better of the discussion so far.

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Your challenge is one calculation, and I'll walk you through it.

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Find out how much electricity your household uses in a year. It'll be on a bill. A typical figure is somewhere around ten thousand kilowatt-hours. Now the question is: what mass, converted entirely to energy, would supply that?

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One kilowatt-hour is three point six million joules. So ten thousand of them is thirty-six billion joules. Divide by c squared, that nine with sixteen zeros.

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You get four ten-millionths of a kilogram. Less than half a milligram.

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A year of light, heat, cooking, washing, and this podcast: the mass of a grain of sand. That's the exchange rate.

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Next time. Einstein has dealt with steady motion. But there's something his theory can't handle, and it's rather a large omission. Gravity. It takes him ten more years, it nearly breaks him, and it begins with what he called the happiest thought of his life: a man falling off a roof.