Episode 25 - Gravity is geometry

Tomlin Hawthorne

You're in a lift, at the top of a very tall building, holding an apple. The cable snaps.

Tomlin Hawthorne

Let's say, for the sake of your peace of mind, that there's an excellent safety system at the bottom, and you'll be fine.

Tomlin Hawthorne

While you're falling, you let go of the apple.

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What does it do?

Tomlin Hawthorne

Nothing. It stays exactly where you let go of it. It floats in front of your face. Because it's falling at the same rate as you are, and the lift is, and the coins that have just drifted up out of your pocket.

Tomlin Hawthorne

Inside that falling lift, there is no gravity. You're weightless. If the lift had no windows, you'd have no way to tell you weren't floating in deep space, a billion miles from anywhere.

Tomlin Hawthorne

In 1907 a man sitting at a desk in a patent office imagined that, and said later that it was the happiest thought of his life. Because he saw that it was a clue to what gravity is. And the answer, which took him eight more years, is that gravity isn't a force at all.

Tomlin Hawthorne

Last time you worked out how much mass, turned wholly into energy, would run your household for a year. The answer was under half a milligram. A grain of sand. That's what c squared does.

Tomlin Hawthorne

So: by 1905 Einstein has a theory of space, time and motion. But it only covers steady motion, in a straight line. And there's an embarrassing gap in it. Newton's gravity doesn't fit.

Tomlin Hawthorne

In Newton's theory, the Sun pulls on the Earth instantly, across ninety-three million miles. If the Sun moved, the Earth would feel it at once. But Einstein has just shown that nothing can travel faster than light, and that at once has no universal meaning anyway. So Newton's gravity has to go, or be rebuilt. Today's question is the oldest one in the series. What is gravity?

Tomlin Hawthorne

Let me begin at the end, because it's one of the great scenes.

Tomlin Hawthorne

London, the sixth of November, 1919. A joint meeting of the Royal Society and the Royal Astronomical Society. The room is packed. On the wall hangs a portrait of Isaac Newton, who was once the Society's president. The mathematician Alfred North Whitehead was there, and wrote that the whole atmosphere was like a Greek drama. They'd come to hear whether a prediction made by a German, in the middle of the war, about the bending of starlight, had been borne out by a British expedition.

Tomlin Hawthorne

It had. And the next morning The Times ran the headline: Revolution in Science. New Theory of the Universe. Newtonian Ideas Overthrown.

Tomlin Hawthorne

Within a week, Einstein, who was forty and known only to physicists, was the most famous person on Earth. He stayed that way for the rest of his life. We'll come back to that room. First, the thought.

Tomlin Hawthorne

Einstein told the story himself. I was sitting in a chair in the patent office in Bern, he said, when all of a sudden a thought occurred to me. If a person falls freely, he won't feel his own weight. I was startled.

Tomlin Hawthorne

It's the lift. And you know it's true, because you saw it in episode five. The astronauts on the space station float because they're falling, all the time, round the Earth. Gravity up there is nearly as strong as it is down here. They don't feel it, because they're not resisting it.

Tomlin Hawthorne

Now think about what you feel right now as weight. You're pressed into your chair. But is that gravity you're feeling? What you feel is the chair pushing up on you. If the chair vanished, and the floor, and you fell, the feeling of weight would vanish too.

Tomlin Hawthorne

So here's Einstein's first move. He turns the lift thought round.

Tomlin Hawthorne

Put yourself in a sealed box with no windows, far out in space, away from any planet. You float. Now attach a rocket to the box and switch it on, so that the box accelerates steadily, in the direction of what you'll soon be calling up.

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The floor comes up and meets your feet. You're pressed against it. Let go of an apple, and the floor rushes up to meet the apple: from inside, the apple falls. Everything falls, and everything falls at the same rate, feathers and hammers alike, for the simple reason that it's really the floor that's moving.

Tomlin Hawthorne

From inside that box, it's exactly like standing in a room on Earth.

Tomlin Hawthorne

And Einstein said: not just like. The same. There is no experiment you can do, inside the box, to tell whether you're being accelerated by a rocket, or sitting still in a gravitational field. He called this the principle of equivalence.

Tomlin Hawthorne

You'll notice it has the same shape as his other principles. Galileo said: you can't tell steady motion from rest. Einstein says: you can't tell acceleration from gravity. He keeps finding things you can't tell apart, and insisting that therefore they're the same.

Tomlin Hawthorne

And it explains at a stroke the thing Galileo found and nobody had ever explained: why all bodies fall at the same rate. For Newton that was an odd coincidence. For Einstein it's obvious. They're not being pulled. The floor is coming up.

Tomlin Hawthorne

Now watch him use it.

Tomlin Hawthorne

You're back in the rocket box, accelerating. On one wall there's a torch, and it sends a pulse of light straight across the box to the opposite wall.

Tomlin Hawthorne

Light travels in a straight line. But while that pulse is crossing the box, the box is accelerating upward. The far wall has moved up by the time the light gets there. So the light hits the far wall a little lower than the height it started from.

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From inside the box, what did you see? You saw the light beam curve. It drooped, very slightly, like a thrown ball.

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And now the principle of equivalence. If light curves in an accelerating box, and an accelerating box can't be told apart from gravity, then light must curve in gravity too.

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But hold on. Light is the very definition of a straight line. When a surveyor wants a straight line, she uses a beam of light. When you check whether a plank is straight, you sight along it. If light doesn't go straight near a heavy object, then nothing does. There's no straighter thing to compare it with.

Tomlin Hawthorne

So Einstein draws the conclusion. Near a massive object, there are no straight lines in the usual sense. Space itself is curved. And not just space. Spacetime, the four-dimensional thing from two episodes ago.

Tomlin Hawthorne

It took him eight years to turn that into a theory. He had to learn a whole branch of mathematics he'd skipped as a student, the geometry of curved spaces, with the help of an old classmate. He wrote to a colleague that he'd never worked so hard in his life, and that compared with this problem, the original relativity was child's play. In the final month, November 1915, he was racing David Hilbert, the great mathematician from our Noether episode, who'd got interested. Einstein gave four lectures on four successive Thursdays in Berlin, correcting himself as he went, and reached the final equations in the last one.

Tomlin Hawthorne

Here's what they say. And I can't improve on the summary given by John Wheeler, the American physicist. Spacetime tells matter how to move. Matter tells spacetime how to curve.

Tomlin Hawthorne

That's the whole theory. Mass and energy bend the spacetime around them. And things moving freely through that bent spacetime follow the straightest paths available.

Tomlin Hawthorne

There's no force. The Earth isn't being pulled by the Sun. The Sun has put a dent in spacetime, and the Earth is going as straight as it can through the dent. And the straightest path through curved spacetime looks, to us, like an orbit.

Tomlin Hawthorne

Let me give you a picture that I think is honest, and then one that isn't.

Tomlin Hawthorne

The honest one. Two explorers stand on the equator, a hundred miles apart. They both set off due north, at the same speed. They're walking on parallel tracks. Neither of them ever turns.

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And yet they get closer. Slowly at first, then faster. And at the North Pole, they collide.

Tomlin Hawthorne

If they didn't know the Earth was round, they'd be baffled. They walked perfectly straight. What pulled them together? They might well conclude there was a force, an attraction between explorers.

Tomlin Hawthorne

There's no force. They each walked in the straightest possible line. It's just that they were walking on a curved surface, and on a curved surface, straight lines that start out parallel can meet.

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That's gravity. You and the Earth are both travelling through spacetime as straight as you can. Spacetime is curved, so your paths converge. And we've been calling that convergence a force for three hundred years.

Tomlin Hawthorne

Now the other picture. You've seen it in every documentary. A stretched rubber sheet, with a bowling ball in the middle making a dip, and a marble rolling round the dip.

Tomlin Hawthorne

It's not a bad start. It shows mass curving something, and it shows orbits coming out of the curve. But ask yourself: why does the marble roll down into the dip? Because gravity is pulling it down. The real gravity, of the room the demonstration's in. So the picture is using gravity to explain gravity. Do it on a space station and the marble would just sit there.

Tomlin Hawthorne

And it leaves out the more important half. It shows space bending. But for ordinary things, the Earth, an apple, you, it's mostly the bending of time that does the work.

Tomlin Hawthorne

Here's what that means. Time runs slower the closer you are to a mass. A clock on the floor ticks more slowly than a clock on the ceiling.

Tomlin Hawthorne

This was first measured in 1959, by two physicists at Harvard, Robert Pound and Glen Rebka, in a tower twenty-two metres high. They sent a beam from the top to the bottom and showed that time at the bottom runs slow by a couple of parts in a thousand million million. Today's best atomic clocks can detect the difference when you raise one of them by a single millimetre.

Tomlin Hawthorne

So your head is ageing faster than your feet. Over a lifetime, by a little under a millionth of a second. People who live in the mountains age faster than people who live by the sea. It isn't much of a reason to move.

Tomlin Hawthorne

And remember the satellites. I said there were two corrections. Their speed makes their clocks run slow, by seven millionths of a second a day. But they're high up, further from the Earth, where time runs faster, by forty-five millionths of a second a day. The net effect is thirty-eight millionths, fast. The engineers who built the first navigation satellite weren't all convinced this was real. So they launched it with the correction built in, but switched off, and a way to turn it on from the ground. After twenty days they turned it on.

Tomlin Hawthorne

Now, a theory this strange has to earn its keep. It did, three times.

Tomlin Hawthorne

The first was a small old mystery. The planet Mercury goes round the Sun in an oval, and the oval itself slowly swivels. Newton's gravity, allowing for the tugs of the other planets, explained nearly all of the swivel. But there was a tiny bit left over, about a hundredth of a degree per century, that nobody could account for. You'll remember from episode five that when Uranus misbehaved, they found Neptune. So astronomers assumed there must be another planet, inside Mercury's orbit. They even named it. Vulcan. Several people claimed to have seen it. It isn't there.

Tomlin Hawthorne

In November 1915, Einstein fed Mercury into his new equations. With no adjustment, no fudge, out came the missing swivel. Exactly. He said afterwards that he was beside himself for days. He felt his heart thumping, as if something inside him had snapped.

Tomlin Hawthorne

The second was a new prediction. Starlight passing close to the Sun should be bent. Stars near the edge of the Sun should appear slightly out of place. Of course, you can't see stars near the Sun, except for a few minutes during a total eclipse.

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The man who went to look was Arthur Eddington, a Cambridge astronomer. He was a Quaker, and a pacifist, and in 1918 he was about to be sent to a labour camp for refusing to fight. His colleagues got him deferred on one condition: that if the war ended in time, he'd lead an expedition to observe the eclipse of May 1919. So an English conscientious objector sailed to a small island off the west coast of Africa to test the theory of a scientist in Berlin, within months of the Armistice. Both of them saw it as an act of repair.

Tomlin Hawthorne

It rained that morning. The clouds thinned just enough, just in time. He got a handful of usable photographs. A second team, in Brazil, got more. The stars had shifted. And by the amount Einstein predicted, which was twice what Newton's theory would give.

Tomlin Hawthorne

That was the announcement in the room with Newton's portrait.

Tomlin Hawthorne

There's a story from that autumn. A student was with Einstein when a telegram arrived with the news. He handed it to her, quite calmly. She said how wonderful, and asked what he'd have done if the result had gone the other way. He said: then I'd have been sorry for the good Lord. The theory is correct.

Tomlin Hawthorne

And look at what's been answered. In episode five, Newton himself said that gravity acting across empty space, with nothing in between, was an absurdity. He couldn't fix it. Einstein has. Nothing acts at a distance. Matter bends the spacetime where it is. The bend spreads outward, at the speed of light. And a distant object responds to the shape of spacetime where it is. It's Faraday's idea, the field, applied to gravity. Only this time, the field is space and time themselves.

Tomlin Hawthorne

The third test I'll keep for next week. Because the equations also said that spacetime can ripple. And that if you bend it far enough, it closes up.

Tomlin Hawthorne

The myth of the week is in the rubber sheet: that gravity pulls things down a slope.

Tomlin Hawthorne

Nothing is pulled. Falling is what things do when nothing is acting on them. It's the natural, force-free state. The astronaut is the one having the true experience. What needs explaining isn't why the apple falls. It's why you don't. And the answer is that the ground is in your way, pushing you off the path you'd otherwise take. The force you feel in your feet is the floor, preventing you from moving naturally through spacetime.

Tomlin Hawthorne

And the question to take away.

Tomlin Hawthorne

The philosopher Karl Popper was seventeen in 1919, in Vienna, and the eclipse made a great impression on him. Here was a theory that had stuck its neck out. It said: the light will bend by this much. If it had bent by half that, or not at all, the theory was dead. Compare that, he said, with the theories of his day about history or the mind, which seemed able to explain anything whatever that happened. He decided that this was the mark of real science: that it takes risks. That it says what would prove it wrong.

Tomlin Hawthorne

But later historians looked hard at Eddington's photographs, and some of them asked whether the data were really as clear as he'd claimed. Whether he'd wanted the answer too much. More recent work, by Daniel Kennefick, concludes that he did it properly. And the measurement's been repeated many times since, far more accurately, and Einstein was right.

Tomlin Hawthorne

Still. Does it matter that the famous test was messier than the legend? Does a theory stand or fall by one experiment? And how much should it count that the man doing the measuring badly wanted it to work?

Tomlin Hawthorne

Your challenge. You need an orange and a pen.

Tomlin Hawthorne

Draw a triangle on the orange. A big one. Make each side as straight as you can: the shortest path along the surface from one corner to the next. A good choice is one corner at the top, where the stalk is, and two on the widest part, a quarter of the way round from each other.

Tomlin Hawthorne

Now look at the angles at the corners. On a flat sheet of paper, the three angles of any triangle add up to a hundred and eighty degrees. That's Euclid. On your orange, each of those three angles is a right angle. Ninety, ninety, ninety. Two hundred and seventy.

Tomlin Hawthorne

That's how you'd know you lived on a curved surface, without ever leaving it. And it's how we know spacetime is curved.

Tomlin Hawthorne

Next time. What happens if you keep adding mass, and keep bending spacetime, until it can't bend any further? In 1916, a German officer on the Russian front worked out the answer in a letter to Einstein. He was dead within months. What he'd found was so strange that Einstein spent the rest of his life hoping it wasn't real.