Science of Discworld III

by Terry Pratchett

  ‘I hope you know what you’re doing, Stibbons,’ hissed Ridcully as the boy scurrièd away.

  ‘Totally, Archchancellor. Hex says the line of causality is – ah, Mrs Boddy?’

  This was to a skinny, worried woman who was advancing on them from the dim interior, wiping her hands on her apron.

  ‘I am, sir,’ said the cook. ‘The boy said you gentlemen was Hygienic?’

  ‘Mrs Boddy, you had some fish delivered this morning?’ said Ponder, sternly.

  ‘Yes sir. Nice piece o’ hake.’ Sudden uncertainty seized her features. ‘Er … that was all right, wasn’t it?’

  ‘Alas it was not, Mrs Boddy!’ said Ponder. ‘We have just come from the fishmonger. All his hake is completely off. We have had many complaints. Some of them were from next of kin, Mrs Boddy!’

  ‘Oh, what shall we do to be saved!’ the cook burst out. ‘I’ve got it cookin’! It smelled all right, sir!’

  ‘Thankfully, then, there is no harm done,’ said Ponder.

  ‘Shall I give it to the cat?’

  ‘Do you like the cat’ said Ponder. ‘No, wrap it in some paper and bring it out to us right now! I’m sure Mr Souser will understand when you give him some of the cold ham from yesterday.’

  ‘Yessir!’ The cook scurried away, and returned shortly with a parcel of very hot, very damp fish. Ponder grabbed it from her and thrust it into Rincewind’s arms.

  ‘Scour the pan thoroughly, Mrs Boddy!’ said Ponder, as Rincewind tried to juggle hake. ‘Gentlemen, we must hurry!’

  He started to walk very fast towards the end of the street, the wizards jogging along behind him, and turned sharply into an alleyway just ahead of a shout of ‘Sir? Sir? How did you know about the cold ham?’

  ‘Location 9, Hex,’ said Ponder. ‘And remove the fish, please!’

  ‘Was all that about?’ said Ridcully. ‘Why did we take that poor woman’s fish?’

  Rincewind said ‘ow!’ as the fish disappeared.

  ‘Mr Souser will travel, er, tomorrow to meet some businessmen,’ said Ponder, as a circle formed on the ground around the wizards. ‘One of them will be a man called Josiah Wedgwood, a famous industrialist. Mr Souser will tell him about his son James, who is currently working with the Navy. It has made a man of him, Mr Souser will say. Mr Wedgwood will listen with interest, and form the opinion that the adventure of a long sea voyage in respectable company may well be of benefit to a young man on the verge of adult life. At least, he will now. If Mr Souser had eaten that fish, he would have been too ill to travel tomorrow.’

  ‘Well, that’s good news for Mr Souser, but what’s it got to do with us?’ said the Dean.

  ‘Mr Wedgwood is Charles Darwin’s uncle,’ said Ponder, as the air wavered. ‘He will have an influence on his nephew’s career. And now for our next call …’

  ‘Good morning! Mrs Nightingale?’

  ‘Yes?’ said the woman, as if she was now doubting it. She took in the group of people in front of her, her eye resting on the very bearded one whose knuckles touched the ground. Beside her, the housemaid who’d opened the door looked on nervously.

  ‘My name is Mr Stibbons, Mrs Nightingale. I am the secretary of The Mission to Deep Sea Voyagers, a charitable organisation. I believe Mr Nightingale is shortly to embark on a perilous mission to the storm-tossed, current-mazed, ship-eating giant-squid infested waters of the South Americas?’

  The woman’s gaze tore itself away from the Librarian and her eyes narrowed.

  ‘He never said anything to me about giant squid,’ she said.

  ‘Indeed? I’m very sorry to hear that, Mrs Nightingale. Brother Bookmeister here,’ Ponder patted the Librarian on the shoulder, ‘would tell you about them himself were it not that the dire experience quite robbed him of the power of speech.’

  ‘Ook!’ said Brother Bookmeister plaintively.

  ‘Really?’ said the woman, setting her jaw firmly. ‘Would you gentlemen care to step into the parlour?’

  ‘Well, the biscuits were nice,’ said the Dean, as the wizards strolled out into the street half an hour later. ‘And now, Stibbons, would you care to tell us what all that was about?’

  ‘Gladly, Dean, and may I say your story about the sea snake was very useful?’ said Ponder. ‘But Rincewind, that tale about the killer flying fish was rather over the top, I thought.’

  ‘I didn’t make it up!’ Rincewind said. ‘They had teeth on them like—’

  ‘Well, anyway … Darwin was the second choice for the post on the Beagle,’ said Ponder. ‘Mr Nightingale was the captain’s initial choice. History will record that after his wife’s pleading he declined the offer. This he will do within about five minutes of when he gets home tonight.’

  ‘Another fine ruse?’ said Ridcully.

  ‘I’m rather pleased with it, as a matter of fact,’ said Ponder.

  ‘Hmm,’ said Ridcully. Cunning in younger wizards is not automatically applauded in their elders. ‘Very clever, Stibbons. You are a wizard to watch.’

  ‘Thank you, sir. My next question is: does anyone here know anything about shipbuilding? Well, perhaps that won’t be necessary. Hex, take us to Portsmouth, please. The Beagle is being refitted. You will need to be naval inspectors which, ahaha, I’m sure you’ll be good at. In fact you will be the most observant inspectors there have ever been. Location 3, please, Hex.’

  1 Yes, they did – in The Science of Discworld II.

  2 A rare meteorological phenomenon discussed briefly in The Science of Discworld II.

  EIGHT

  FORWARD TO THE PAST

  WELL, THE WIZARDS HAVE MADE a good start. And with the might of Hex behind them, the wizards can travel at will along the Roundworld timeline. We’re happy for them to do that, in a fictional context – but could we do the same thing, in a factual one?

  To answer that, we must decide what a time machine looks like within the framework of general relativity. Then we can talk about building one.

  Travel into the future is easy: wait. It’s getting back that’s hard. A time machine lets a particle or object return to its own past, so its world-line, a timelike curve, must close into a loop. So a time machine is just a closed timelike curve, abbreviated to CTC. Instead of asking, ‘Is time travel possible?’ we ask, ‘Can CTCs exist?’

  In flat Minkowski spacetime, they can’t. Forward and backward light cones – the future and past of an event – never intersect (except at the point itself, which we discount). If you head off across a flat plane, never deviating more than 45° from due north, you can never sneak up on yourself from the south.

  But forward and backward light cones can intersect in other types of spacetime. The first person to notice this was Kurt Gödel, better known for his fundamental work in mathematical logic. In 1949 he worked out the relativistic mathematics of a rotating universe, and discovered that the past and future of every point intersect. Start wherever and whenever you like, travel into your future, and you’ll end up in your own past. However, observations indicate that the universe is not rotating, and spinning up a stationary universe (especially from inside) doesn’t look like a plausible way to make a time machine. Though, if the wizards were to give Roundworld a twirl …

  The simplest example of future meeting past arises if you take Minkowski spacetime and roll it up along the ‘vertical’ time direction to form a cylinder. Then the time coordinate becomes cyclic, as in Hindu mythology, where Brahma recreates the universe every kalpa, a period of 4.32 billion years. Although a cylinder looks curved, the corresponding spacetime is not actually curved – not in the gravitational sense. When you roll up a sheet of paper into a cylinder, it doesn’t distort. You can flatten it out again and the paper is not folded or wrinkled. An ant that is confined purely to the surface won’t notice that spacetime has been bent, because distances on the surface haven’t changed. In short the local metric doesn’t change. What changes is the global geometry of spacetime, its overall topology.

  Rolling up Minkowski
spacetime is an example of a powerful mathematical trick for building new spacetimes out of old ones: cut-and-paste. If you can cut pieces out of known spacetimes, and glue them together without distorting their metrics, then the result is also a possible spacetime. We say ‘distorting the metric’ rather than ‘bending’, for exactly the reason that we say that rolled-up Minkowski spacetime is not curved. We’re talking about intrinsic curvature, as experienced by a creature that lives in the spacetime, not about apparent curvature as seen by some external viewer.

  The rolled-up version of Minkowski spacetime is a very simple way to prove that spacetimes that obey the Einstein equations can possess CTCs – and thus that time travel is not inconsistent with currently known physics. But that doesn’t imply that time travel is possible. There is a very important distinction between what is mathematically possible and what is physically feasible.

  A spacetime is mathematically possible if it obeys the Einstein equations. It is physically feasible if it can exist, or could be created, as part of our own universe or an add-on. There’s no very good reason to suppose that rolled-up Minkowski spacetime is physically feasible: certainly it would be hard to refashion the universe in that form if it wasn’t already endowed with cyclic time, and right now very few people (other than Hindus) think that it is. The search for spacetimes that possess CTCs and have plausible physics is a search for more plausible topologies. There are many mathematically possible topologies, but, as with the Irishman giving directions, you can’t get to all of them from here.

  However, you can get to some remarkably interesting ones. All you need is black hole engineering. Oh, and white holes, too. And negative energy. And –

  One step at a time. Black holes first. They were first predicted in classical Newtonian mechanics, where there is no limit to the speed of a moving object. Particles can escape from an attracting mass, however strong its gravitational field, by moving faster than the appropriate ‘escape velocity’. For the Earth, this is 7 miles per second (11 kps), and for the Sun, it is 26 miles per second (41 kps). In an article presented to the Royal Society in 1783, John Michell observed that the concept of escape velocity, combined with a finite speed of light, implies that sufficiently massive objects cannot emit light at all – because the speed of light will be lower than the escape velocity. In 1796 Pierre Simon de Laplace repeated these observations in his Exposition of the System of the World. Both of them imagined that the universe might be littered with huge bodies, bigger than stars, but totally dark.

  They were a century ahead of their time.

  In 1915 Karl Schwarzschild took the first step towards answering the relativistic version of the same question, when he solved the Einstein equations for the gravitational field around a massive sphere in a vacuum. His solution behaved very strangely at a critical distance from the centre of the sphere, now called the Schwarzschild radius. It is equal to the mass of the star, multiplied by the square of the speed of light, multiplied by twice the gravitational constant, if you must know.

  The Schwarzschild radius for the Sun is 1.2 miles (2 km), and for the Earth 0.4 inches (1 cm) – both buried inaccessibly deep where they can’t cause trouble. So it wasn’t entirely clear how significant the strange mathematical behaviour was … or even what it meant.

  What would happen to a star that is so dense that it lies inside its own Schwarzschild radius?

  In 1939 Robert Oppenheimer and Hartland Snyder showed that it would collapse under its own gravitational attraction. Indeed a whole portion of spacetime would collapse to form a region from which no matter, not even light, could escape. This was the birth of an exciting new physical concept. In 1967 John Archibald Wheeler coined the term black hole, and the new concept was christened.

  How does a black hole develop as time passes? An initial clump of matter shrinks to the Schwarzschild radius, and then continues to shrink until, after a finite time, all the mass has collapsed to a single point, called a singularity. From outside, though, we can’t observe the singularity: it lies beyond the ‘event horizon’ at the Schwarzschild radius, which separates the observable region, from which light can escape, and the unobservable region where the light is trapped.

  If you were to watch a black hole collapse from outside, you would see the star shrinking towards the Schwarzschild radius, but you’d never see it get there. As it shrinks, its speed of collapse as seen from outside approaches that of light, and relativistic time-dilation implies that the entire collapse takes infinitely long when seen by an outside observer. The light from the star would shift deeper and deeper into the red end of the spectrum. The name should be ‘red hole’.

  Black holes are ideal for spacetime engineering. You can cut-and-paste a black hole into any universe that has asymptotically flat regions, such as our own.1 This makes black hole topology physically plausible in our universe. Indeed, the scenario of gravitational collapse makes it even more plausible: you just have to start with a big enough concentration of matter, such as a neutron star or the centre of a galaxy. A technologically advanced society could build black holes.

  A black hole doesn’t possess CTCs, though, so we haven’t achieved time travel. Yet. However, we’re getting close. The next step uses the time-reversibility of Einstein’s equations: to every solution there corresponds another that is just the same, except that time runs backwards. The time reversal of a black hole is called a white hole. A black hole’s event horizon is a barrier from which no particle can escape; a white hole’s event horizon is one into which no particle can fall, but from which particles may emerge at any moment. So, seen from the outside, a white hole would appear as the sudden explosion of a star’s worth of matter, coming from a time-reversed event horizon.

  White holes may seem rather strange. It makes sense for an initial concentration of matter to collapse, if it is dense enough, and thus to form a black hole; but why should the singularity inside a white hole suddenly decide to spew forth a star, having remained unchanged since the dawn of time? Perhaps because time runs backwards inside a white hole, so causality runs from future to past? Let’s just agree that white holes are a mathematical possibility, and notice that they too are asymptotically flat. So if you knew how to make one, you could glue it neatly into your own universe, too.

  Not only that: you can glue a black hole and a white hole together. Cut them along their event horizons, and paste along these two horizons. The result is a sort of tube. Matter can pass through the tube in one direction only: into the black hole and out of the white. It’s a kind of matter-valve. The passage through the valve follows a timelike curve, because material particles can indeed traverse it.

  Both ends of the tube can be glued into any asymptotically flat region of any spacetime. You could glue one end into our universe, and the other end into somebody else’s; or you could glue both ends into ours – anywhere you like except near a concentration of matter. Now you’ve got a wormhole. The distance through the wormhole is very short, whereas that between the two openings, across normal spacetime, can be as big as you like.

  A wormhole is a short cut through the universe.

  But that’s matter-transmission, not time travel.

  Never mind: we’re nearly there.

  The key to wormhole time travel is the notorious twin paradox, pointed out by the physicist Paul Langevin in 1911. Recall that in relativity, time passes more slowly the faster you go, and stops altogether at the speed of light. This effect is known as time dilation. We quote from The Science of Discworld:

  Suppose that Rosencrantz and Guildenstern are born on Earth on the same day. Rosencrantz stays there all his life, while Guildenstern travels away at nearly lightspeed, and then turns round and comes home again. Because of time dilation, only one year (say) has passed for Guildenstern, whereas 40 years have gone by for Rosencrantz. So Guildenstern is now 39 years younger than his twin brother.

  It’s called a paradox because there seems to be a puzzle: from Guildenstern’s frame of reference, it i
s Rosencrantz who has whizzed off at near-lightspeed. Surely, by the same token, Rosencrantz should be 39 years younger, not Guildenstern? But the apparent symmetry is fallacious. Guildenstern’s frame of reference is subject to acceleration and deceleration, especially when he turns round to head for home; Rosencrantz’s isn’t. In relativity, accelerations make a big difference.

  In 1988 Michael Morris, Kip Thorne, and Ulvi Yurtsever realised that combining a wormhole with the twin paradox yields a CTC. The idea is to leave the white end of the wormhole fixed, and to zigzag the black one back and forth at just below the speed of light. As the black end zigzags, time dilation comes into play, and time passes more slowly for an observer moving with that end. Think about world-lines that join the two wormholes through normal space, so that the time experienced by observers at each end are the same. At first those lines are almost horizontal, so they are not timelike, and it is not possible for material particles to proceed along them. But as time passes, the line gets closer to the vertical, and eventually it becomes timelike. Once this ‘time barrier’ is crossed, you can travel from the white end of the wormhole to the black through normal space – following a timelike carve. Because the wormhole is a short cut, you can do so in a very short period of time, effectively travelling instantly from the black end to the corresponding white one. This is the same place as your starting point, but in the past.

  You’ve travelled in time.

  By waiting, you can close the path into a CTC and end up at the same place and time that you started from. Not back to the future, but forward to the past. The further into the future your starting point is, the further back in time you can travel from that point. But there’s one disadvantage of this method: you can never travel back past the time barrier, and that occurs some time after you build the wormholes. No hope of going back to hunt dinosaurs. Or to tread on Cretaceous butterflies.