Episode 07 - Symmetry: Why anything is conserved

Tomlin Hawthorne

Imagine you do an experiment today. Anything you like. You drop a ball and time it. Now imagine doing exactly the same experiment next Tuesday. Same ball, same height, same everything.

Tomlin Hawthorne

Of course you will. It's barely a question. Why would Tuesday be different? Hold on to how obvious that felt. Because by the end of this episode I'm going to show you that this dull little fact, that the rules are the same on Tuesday, is the reason energy can't be created or destroyed. They're the same fact. And the person who proved it is, I think, the most important scientist most people have never heard

Tomlin Hawthorne

of.Last time, a bouncing ball. Each bounce a little lower, and I asked where the energy goes. If you listened, you heard part of the answer: every bounce makes a sound, and sound is energy leaving. The rest goes into heat. The ball and the floor both end up a touch warmer. The ledger balances. It always does. And your electricity bill in cookies. A typical household comes out at several thousand a month. I hope that put the cookie in a new light. So we finished with a law. Energy is conserved. And I told you it has never once been seen to fail. But that should leave you a bit unsatisfied. Why is it conserved? A law that's just true, with no reason, is a strange thing to build a science on. Today's question: where do the conservation laws come from?

Tomlin Hawthorne

To answer it I have to take you to a German university town called Göttingen, in 1915. Göttingen at that time was the centre of the mathematical world. Its presiding figure was David Hilbert, probably the greatest mathematician alive. And Hilbert had become fascinated by a new theory of gravity that a man named Einstein was just then finishing. But there was a puzzle in it, a nasty one. In Einstein's theory, energy didn't seem to be conserved in the ordinary way. Hilbert couldn't crack it. So he sent for help. The person he sent for was Emmy Noether. She was thirty-three. She'd grown up in the university town of Erlangen, the daughter of a mathematics professor, at a time when women in Germany weren't permitted to enrol at university. She'd been allowed to sit in on lectures, with each professor's individual permission, as a kind of tolerated guest. When the rules finally changed she took her doctorate, with the highest distinction, and then worked at the university for seven years without a title and without a salary. Hilbert wanted her appointed as a lecturer at Göttingen. The mathematicians agreed. But the appointment had to go through the wider faculty, the philosophers and the historians, and they were appalled. One of them is supposed to have asked: what will our soldiers think, when they come back from the front and find they're expected to learn at the feet of a woman? And Hilbert, who was not a patient man, is said to have replied: gentlemen, I do not see that the sex of the candidate is an argument against her. We are a university, not a bathing establishment. He lost the vote. So for the next four years, the lecture list at Göttingen advertised courses by Professor Hilbert, and in smaller print, with the assistance of Dr Noether. She gave the lectures. She wasn't paid. And in the middle of this, in 1918, she solved Hilbert's problem. But she did it the way great mathematicians do. She didn't just answer the question. She found the general truth it was a special case of. And that general truth is what we now call Noether's theorem.

Tomlin Hawthorne

The idea is symmetry, and I need to change what you mean by the word. In everyday speech, symmetry means the left side matches the right. A face, a butterfly, the front of a Palladian villa. Physicists mean something broader, and I think more beautiful. A thing has a symmetry if there's something you can do to it that leaves it looking the same. Take a perfectly smooth white ball. Close your eyes. I turn the ball by some amount. Open your eyes. Can you tell I did anything? No. That's a symmetry: the ball is unchanged by rotation. Now a square tile. If I turn it by a quarter turn while you're not looking, you can't tell. If I turn it by a tenth of a turn, you can. So a square has less symmetry than a ball. It survives some changes and not others. Or think of wallpaper with a repeating pattern. Slide it along by exactly one repeat and it sits just where it was.

Tomlin Hawthorne

So: a symmetry is a change that makes no difference. It's a kind of indifference. Now here's the step. Objects can have symmetries. But so can laws. Go back to my opening question. You do an experiment today, and again next Tuesday, and get the same answer. What that says is: the laws of physics are indifferent to when. Slide the whole experiment along in time, and nothing changes. The laws have a symmetry, just like the wallpaper. And there's another. Do the experiment in your kitchen, then carry everything to Paris and do it again. Same result. The laws are indifferent to where. And a third. Do it facing north, then turn the whole apparatus to face east. Same result. The laws don't care about direction. Space has no preferred way up. Anaximander saw that in episode one. These all sound too obvious to mention. Noether's theorem says they're not. Here it is. For every such symmetry of the laws, there is a quantity that is conserved. One for one. Every indifference comes with something that can't change. The laws don't care when. And that is why energy is conserved. The laws don't care where. And that is why something called momentum is conserved. Momentum is the quantity of motion, the thing that makes a recoiling gun kick back. The laws don't care which direction. And that is why spin is conserved, what physicists call angular momentum. You've seen it. A figure skater spinning with her arms out pulls them in and whirls faster. She has to. The amount of spin has to stay the same, and with her mass closer in, the only way to keep it is to turn quicker.

Tomlin Hawthorne

I'd like to try to make one of these feel reasonable, because "it's a theorem" is not very satisfying on a podcast. Take energy and time. Suppose the laws did change with time. Suppose gravity were a little weaker on Mondays and a little stronger on Fridays. Then I could make a fortune. On Monday, when gravity's weak, I pump a great deal of water up a hill. It's cheap, the water's light. Then I wait. On Friday, when gravity's strong, I let it run back down through a turbine. It's heavier now, so I get out more energy than I put in. I do it again next week. Energy from nothing, every week, for ever. So if the laws changed with time, energy would not be conserved. You could mine the difference. Turn that round, and you have Noether: if the laws are the same at all times, there is no difference to mine. The books have to balance. The ledger from last episode balances because the rules of accounting don't change overnight. I think that's one of the most satisfying things I know. The conservation of energy looked like a brute fact about the world. It turns out to be the shadow of a kind of sameness, the sameness of the laws from one moment to the next. And it changed how physics is done. Before Noether, you discovered a conservation law and felt pleased. After her, physicists do it the other way round. They ask: what symmetries does nature have? And then they let the symmetries dictate the laws. Almost everything in the second half of this series, the forces inside the atom, the particles, the Higgs field, was found by people asking about symmetry first. She handed them the method.

Tomlin Hawthorne

I should finish her story. She did, in the end, get a modest title and a small salary. She went on to rebuild a whole branch of mathematics, abstract algebra, and a generation of the best young mathematicians in Europe trailed after her through the streets of Göttingen. They were known as the Noether boys. By all accounts she was loud, warm, untidy, completely indifferent to appearances, and she talked mathematics so fast that her hair came unpinned. In April 1933, three months after Hitler took power, she was dismissed, along with Jewish professors across Germany. She went to America, to Bryn Mawr, a women's college in Pennsylvania. Two years later she died suddenly after an operation. She was fifty-three. The New York Times ran a short notice. And Einstein, feeling that it didn't begin to do her justice, wrote a letter to the editor. He said that in the judgement of the most competent living mathematicians, she was the most significant creative mathematical genius produced since the higher education of women began. Note the careful compliment, by the way. Even Einstein hedged it. Several of her colleagues thought the qualification was unnecessary.

Tomlin Hawthorne

One more idea, and it's a strange one. I'm planting it now because we'll need it in thirty episodes' time. Sometimes the laws have a symmetry and the world doesn't. Balance a pencil perfectly on its point. Everything about the situation is symmetrical. There's no direction it prefers. And yet it can't stay there. It has to fall, and when it falls, it falls some particular way. Toward the window, say. The law had no favourite direction. The outcome does. A Pakistani physicist called Abdus Salam had a lovely way of putting this. Picture a round dinner table, fully laid, with a napkin between each pair of guests. Is your napkin the one on your left or on your right? There's nothing to choose between them. The table is perfectly symmetrical. And then one guest picks up a napkin, the one on her left, say. And at that moment the question is settled for everyone. The whole table has to go left. That's called a broken symmetry. A symmetrical rule, an unsymmetrical result, and the result is an accident that then becomes binding. Keep the dinner table in mind. It turns out the entire universe did something like this in its first instant, and it's the reason anything has mass.

Tomlin Hawthorne

The myth of the week is small but worth clearing away: that symmetry means mirror images. In physics the symmetries that do the heavy lifting aren't reflections. They're smooth changes. Sliding, turning, waiting. The fact that you can shift by any amount, not just flip. And the question to take with you. It's one physicists argue about with real heat at the moment. Paul Dirac, one of the great physicists of the last century and a man we'll meet properly later, said that it's more important to have beauty in one's equations than to have them fit experiment. He meant it. He thought that if a theory was beautiful enough, it was probably right, and the experiment was probably wrong. He was sometimes vindicated, which is the annoying thing. And by beauty, physicists nearly always mean symmetry. Noether showed them how powerful it is, and for a hundred years they've chased it. But there's a dissenting view. A German physicist, Sabine Hossenfelder, wrote a book a few years ago called Lost in Math. Her argument is that this faith has gone bad. That for forty years, theorists have been building ever more elegant, symmetrical theories that have predicted nothing that's been found. That beauty is a human taste, shaped by what worked last time, and nature is under no obligation to share it. So who's right? Is beauty a guide to truth? You have some standing here. You know what it is to judge a poem, or a proof in philosophy, or the design of a building. Is the sense that something is right because it's elegant a kind of knowledge, or is it a prejudice with good manners? Your challenge. You need a swivel chair and two heavy books. Sit in the chair, hold a book in each hand with your arms stretched straight out to the sides, and get someone to give you a gentle spin. Or push off with a foot. Then pull the books in to your chest. You'll speed up, quite sharply. Push them out again and you'll slow down. That's Noether's theorem in your office. The universe doesn't care which way you're facing, and so your spin is conserved, and you can feel it being conserved in your arms. Please don't do this near anything breakable, and please don't do it straight after lunch. And if you'd like something quieter: take a form you know well. A sonnet, a fugue, a tragedy in five acts. Ask what you could change about it and have it still be the same form. What are its symmetries? What's the one change that would break it? Next time. A beam of light goes from a lamp to your eye, and somehow, out of all the paths it might take, it takes the quickest one. How does it know? The question ended a friendship, gave us the most famous comic novel of the eighteenth century, and still isn't quite comfortable.