Episode 23 - Spacetime

Tomlin Hawthorne

You're sitting still. Or standing, or lying down. Let's say you're not going anywhere.

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How fast are you moving?

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I know. Relative to what? You've learned to ask that. Good.

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But there's an answer that doesn't need a relative to. It's one of the strangest and loveliest ideas in physics, and by the end of this episode I hope it'll seem almost natural.

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You are moving at the speed of light. Right now. So am I. So is everything. We always have been, and it never varies. It's just that at the moment, nearly all of your motion is in a direction you can't point to.

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Last time you drew a light clock, two mirrors and a bouncing pulse, once as seen from the train and once from the platform. A straight line and a zigzag. The zigzag's longer, light's speed is the same for all, so the moving clock ticks slow. And perhaps you explained it to somebody. I hope they're still speaking to you.

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We found that observers moving differently disagree about how much time passed between two events. They disagree about lengths. They even disagree about whether two things happened at the same moment.

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That's a lot of disagreement. It starts to sound as though nothing is solid. So today's question: if space and time are each a matter of perspective, what's left that everyone agrees on?

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Cologne, the twenty-first of September, 1908. A meeting of German scientists and doctors. The speaker is Hermann Minkowski, professor of mathematics at Göttingen.

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Minkowski had taught Einstein in Zurich, ten years earlier, and had not been impressed. He's the one who called him a lazy dog. When the relativity paper came out, Minkowski is said to have told a colleague that he'd never have thought Einstein capable of it.

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But he read it. And he saw something in it that Einstein himself hadn't. Einstein had written the theory in terms of clocks and rulers and trains. Minkowski saw that underneath there was geometry. A shape.

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He opened his lecture with one of the most famous announcements in science. The gist of it was this. From now on, space on its own and time on its own are fated to fade away into shadows. Only a kind of union of the two will keep an independent existence.

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Einstein, at first, thought this was showing off. He grumbled that now the mathematicians had got hold of relativity, he didn't understand it himself any more. Within a few years he realised he couldn't take another step without it.

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Minkowski didn't see what came of it. He died of a burst appendix four months after the lecture, at forty-four. He's supposed to have said, in hospital, what a pity it was to have to die just as relativity was being developed.

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So what did he see?

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I'm going to give it to you through a parable, which comes from a textbook by two American physicists, Edwin Taylor and John Wheeler. It's the best teaching story I know in this whole subject.

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Imagine a kingdom where, for religious reasons, distances north and south are sacred, and are measured in sacred units, miles. Distances east and west are profane, and are measured in feet. Nobody thinks of them as the same kind of thing at all.

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In this kingdom there are two surveyors. One works by day, and takes north from a magnetic compass. The other works by night, and takes north from the pole star. And those two norths aren't quite the same direction. They differ by a small angle.

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Both surveyors measure the position of the town gate relative to the temple. And they get different answers. The day surveyor says the gate is so many miles north and so many feet east. The night surveyor gets a slightly different number of miles and a different number of feet. They argue. Each is sure the other is careless.

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Then a young student has an outrageous idea. She says: what if we measure north and east in the same units? Convert the miles to feet. And then, for each surveyor, take the northward distance squared, add the eastward distance squared, and take the square root.

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She tries it. And the two surveyors, who disagreed on both numbers, get exactly the same answer.

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You know what she's found, because you learned it at school. She's found the distance. The straight-line distance from the temple to the gate. It was there all along. North-ness and east-ness aren't properties of the gate. They depend on which way you've decided is north. But the distance doesn't depend on anything. It's real.

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And once she's seen that, she sees the rest. North and east aren't two different kinds of thing. They're two directions in one thing. A plane. The surveyors were just looking at it from slightly different angles.

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Now here's Minkowski.

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Take two events. A firework goes off here; a moment later, another goes off over there. One observer measures how far apart they were in space, and how far apart in time. A second observer, moving relative to the first, measures a different distance and a different time. They disagree on both, as we saw last week.

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But measure time and space in the same units. And there is a natural conversion: the speed of light. One second of time corresponds to three hundred thousand kilometres of space. Then combine the two numbers, in a way very like the surveyor's sum.

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And both observers get the same answer. Always. However they're moving.

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That combined quantity is called the spacetime interval. It's the true separation between two events. And it's absolute. It's the thing everyone agrees on.

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So the moral is the same as in the kingdom. Space and time aren't two different kinds of thing. They're two directions in one thing, a four-dimensional thing: three of space and one of time. He called it spacetime. And observers in relative motion are like the two surveyors. They're slicing the same reality at different angles. What one of them calls pure time, the other sees as mostly time and a little space.

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Moving fast doesn't distort reality. It rotates your point of view.

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I need to flag one difference, because it's where the analogy breaks, and as always the break is where the physics lives. In the surveyor's sum, you add the two squares. In Minkowski's, you subtract. The time part and the space part come in with opposite signs. That minus sign is the whole difference between time and space. It's why you can turn round in space and you can't turn round in time. It's why there's a speed limit. Everything peculiar about relativity is in that one minus sign.

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Now I can pay off the opening.

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The physicist Brian Greene has a way of putting this that I think is wonderful. Think of a car driving across a huge flat car park at a fixed speed, sixty miles an hour. If it drives due north, all its sixty is going into northward progress. If it turns to drive north-east, it's still doing sixty, but now it's sharing that between north and east. It makes less northward progress, because some of its motion has been diverted.

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Now. Everything in the universe is travelling through spacetime at one fixed rate. Always. The same rate for everything. When you're sitting still, you're not moving through space at all. So all of your travel is going into the other direction. Through time. You're ageing at full speed, one second per second.

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When you start to move through space, you divert some of that. You're still travelling through spacetime at the same fixed rate. But now a little of it is going into space, so there's a little less left for time. Your clock runs slow.

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And the faster you move through space, the less is left over for time. That's time dilation, in one picture.

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And light? Light has diverted everything. It's moving through space at the maximum possible rate, and there's nothing left for time at all. For a ray of light, no time passes. The light that left a distant galaxy two billion years ago and landed in a telescope last night made the journey, by its own clock, in no time whatsoever.

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That's also why nothing can go faster than light. You can't divert more than everything.

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So that's the answer to my question. How fast are you moving? At the speed of light. Through time, mostly.

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Let me give you the picture physicists actually draw.

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Take a sheet of paper. Let left and right stand for position in space. Let up the page stand for time. The future's at the top.

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Now you, sitting in your chair, are not a dot on this page. You're a line. Because you exist at every moment. You're a vertical line, going straight up: same place, later and later. If you get up and walk across the room, your line slants over. Physicists call it your world-line. It's your whole life, drawn as a single thread in spacetime. Birth at the bottom.

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Now, at one moment, switch on a lamp. Light spreads out from it in all directions, at the fixed speed. On the page, that's two lines going up and outward from that point, like a letter V. In the full picture, with more dimensions of space, it's a cone. It's called the light cone.

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And the light cone divides up the universe.

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Inside the V, above you, is every event that a signal from you, here and now, could possibly reach. That's your future. It's everything you could ever affect.

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There's another V opening downward: all the events from which a signal could have reached you by now. That's your past. Everything that could have affected you.

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And outside both cones, off to the sides? That's a region with no name in ordinary language. Physicists call it elsewhere. Events there are too far away, too recently, for light to have got from them to you or from you to them. They can't have caused anything here. Nothing here can have caused them. And it's for those events, and only those, that different observers disagree about which came first.

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That's how causality survives. Observers can argue about the order of two events in the elsewhere, where it makes no difference to anything. They never disagree about the order of a cause and its effect. And that's the deep reason for the speed limit: if anything could outrun light, some observer would see the effect arrive before the cause.

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Here's a consequence you can use tonight. When you look out, you look back. The Sun you see is the Sun of eight minutes ago. If it had vanished seven minutes ago, you'd have no way to know. It'd still be shining in the sky, and the Earth would still be circling the spot where it was. The stars are years old, or centuries. And if you're somewhere dark, in autumn, and can find the faint smudge of the Andromeda galaxy, the light going into your eye left it two and a half million years ago, before there were human beings. The night sky isn't a view. It's an archive.

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And now the hard question.

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If spacetime is the real thing, a four-dimensional whole, and different observers just slice it differently into space and time, then there's no one slice that is the present. Your now and mine tilt apart.

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So think about some event that's in your future. Hasn't happened yet. For someone else, moving differently, far away, that same event can be in their present slice, or their past. For them it's already as real as anything is.

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A lot of physicists and philosophers draw the conclusion that seems to follow. All of it is there. Past, present and future all exist, equally, laid out in spacetime like places on a map. Nothing comes into being; nothing passes away. The whole of history, yours included, just is. It's called the block universe.

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Kurt Vonnegut put this into Slaughterhouse-Five. His aliens, the Tralfamadorians, can see all of time at once, the way we can see a stretch of mountains. So a death doesn't trouble them much. To them a dead person is only dead at one point along the range, and alive and well at all the others.

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And Einstein, late in his life, took something like this view, and took comfort from it. In March 1955, his oldest friend died. Michele Besso, from the patent office, the man he'd thanked in the paper. Einstein wrote to the family. He said that Besso had now left this strange world a little ahead of him. And that this meant nothing. For those who believe in physics, he said, the separation between past, present and future is only an illusion, however persistent.

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Einstein died a month later.

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The myth of the week comes in two sizes.

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One is that the fourth dimension is something mysterious, a spirit realm, where ghosts live. It's the time on your watch. You've been travelling along it all your life.

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The other is the over-correction: that time is just another dimension of space. It isn't. There's that minus sign. You can walk back to the kitchen. You can't walk back to Tuesday.

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And the question to take away.

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The block universe. If the future's already there, as real as the past, what becomes of choice? Of regret, of hope? Is it a liberation, as it seems to have been for Einstein, or is it unbearable?

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And does it square with episode fourteen? There I told you the difference between past and future is real, that it comes from entropy, from the way the universe began. Here I'm telling you past and future are laid out side by side like the north and south of a map. Can both be true? I think they can. A map can have a real slope on it. But it's worth an argument.

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A footnote for the art historians. You'll have heard that Picasso's cubism, showing several viewpoints at once, was inspired by relativity. The dates are suggestive: the first great cubist painting is 1907. The historian Arthur Miller thinks both men were drawing on the same source, the French mathematician Poincaré, whose popular books were read in the cafés. It's a contested claim. I wouldn't lean on it, but I wouldn't dismiss it either.

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Your challenge. Two things.

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One. Draw yesterday. Time up the page, and across the page, your position, simplified to a single line from home to wherever you went. Draw your world-line through the day. Then add the world-line of someone you met. Where the two lines touch is the meeting. Notice that the meeting is a point everyone would agree on, however they were moving. Events are real. It's only the slicing that varies.

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Two. On the next clear night, go out and find the oldest light you can see with your own eyes. Start with the Moon: just over a second old. A bright planet: perhaps half an hour. The bright stars: tens or hundreds of years. And if you can find Andromeda, two and a half million.

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Next time. In September 1905, a few months after the relativity paper, Einstein sent in a three-page afterthought. Its title was a question. Its conclusion was an equation that everyone on Earth now knows and almost nobody can read. We'll read it.