What we have in mind as an alternative to Barbour's timeless probability mist is not a mist that changes as time passes, however. That would fall foul of the non-Newtonian relation between space and time; different parts of the mist would correspond to different times depending on who observed them. No, we're thinking of the mathematical resolution of the Arrow paradox, via Hamiltonian mechanics. Here, the state of a body is given by two quantities, position and momentum, instead of just position. Momentum is a `hidden variable', observable only through its effect on subsequent positions, whereas position is something we can observe directly. We said: `a body in a given position with zero momentum is not moving at that instant, whereas one in the same position with non-zero momentum is moving, even though instantaneously it stays in the same place'. Momentum encodes the next change of position, and it encodes it now. Its value now is not observable now, but it is (will be) observable. You just have to wait to find out what it was. Momentum is a `hidden variable' that encodes transitions from one position to another.
Can we find an analogue of momentum in quantum snakes and ladders? Yes, we can. It is the overall probability of going from any given square to any other. These `transition probabilities' depend only on the squares concerned, not on the time at which the move
is made, so in Barbour's sense they are `timeless'. But when you are on some given square, the transition probabilities tell you where your next move can lead, so you can reconstruct the possible sequences of moves, thereby putting time back into the physics.
For exactly the same reason, a single fixed probability mist is not the only statistical structure with which Platonia can be endowed. Platonia can also be equipped with transition probabilities between pairs of states. The result is to convert Platonia into what statisticians call a `Markov chain', which is just like the list of transition probabilities for snakes and ladders, but more general. If Platonia is made into a Markov chain, each sequence of configurations gets its own probability. The most probable sequences are those that contain large numbers of highly probable states - these look oddly like Barbour's time capsules. So instead of single-state Platonia we get sequentialstate Markovia, where the universe makes transitions through whole sequences of configurations, and the most likely transitions are the ones that provide a coherent history - narrativium.
This Markovian approach offers the prospect of bringing time back into existence in a Platonian universe. In fact, it's very similar to how Susan Sto Helit and Ronnie Soak managed to operate in the cracks between the instants, in Thief of Time.
Tick.
SEVEN
THE FISH IS OFF
Two HOURS LATER A SINGLE sheet of paper slid off Hex's writing table. Ponder picked it up.
`There are about ten points where we must intervene to ensure that The Origin is written,' he said.
'well, that doesn't seem too bad,' said Ridcully. `We got Shakespeare born, didn't we?[1] We just have to tinker.'
`These look a little more complicated,' said Ponder, doubtfully.
`But Hex can move us around,' said Ridcully. `It could be fun, especially if something or someone is playing les buggeurs risibles. It could be educational, Mr Stibbons.'
`And they do really good beer, ` said the Dean. `And the food was excellent. Remember that goose we had last time? I've seldom eaten
better.'
`We will be setting out to save the world,' said Ridcully, severely. `We will have other things on our minds!'
`But there will be mealtimes, yes?' said the Dean
Second Lunch and Mid-afternoon Snack went past almost unnoticed. Perhaps the wizards were already leaving space for goose ...
[1] Yes, they did - in The Science of Discworld II.
It was turning out to be a long day. Easels had been set up around Hex. Paper was strewn across every table. The Librarian had practically built up a branch library in one corner, and was still fetching books from the distant reaches of L-Space.
And the wizards had changed their clothes, ready for hands-on intervention. There had barely been a discussion about it, not after the Dean had mentioned the goose. Hex had a great deal of control over the Globe, but when it came to the fine detail you needed to be hands-on, especially hands on cutlery. Hex had no hands. Besides, he'd explained at length, there was no such thing as absolute control, not in a fully functioning universe. There was just a variable amount of lack of control. In fact, Ponder thought, Hex was a Great Big Thing as far as Roundworld was concerned. Almost ... godlike. But he still couldn't control everything. Even if you knew where every tiny spinning particle of stuff was, you couldn't know what it'd do next.
The wizards would have to go in. They could do that. They'd done it before. No trouble is too much if it saves some excellent chefs from extinction.
Clothing, at least, would not be a problem. Give or take the odd pointy hat and staff, the wizards would be able to walk the Roundworld streets without attracting a second glance.
`How do we look?' said the Archchancellor, as they reassembled.
`Very ... Victorian,' said Ponder. `Although technically, at the moment, very Georgian. Very ... tweedy, anyway. Are you totally happy with the bishop look, Dean?'
`Isn't that appropriate for the time?' said the Dean, looking worried. `We looked through the book on costumes and I thought...' His voice trailed off. `It's the mitre, isn't it ...'
`And the crozier,' said Ponder.
'I wanted to fit in, you see.'
`In a cathedral, yes.. I'm afraid it's plain black suit with gaiters for street wear. However, you can do anything you like with your beard and you can wear hats a small child could stand up inside. But on the streets, bishops are quite dull.'
`Where's the fun in that?' said the Dean, sulkily.
Ponder turned to Rincewind.
`As for you, Rincewind, can I ask why you are wearing nothing but a loincloth and a pointy hat?
'Ah, well, you see, if you don't know what you're getting into, naked always works,' said Rincewind. `It's the all-purpose suit. At home in every culture. Admittedly you sometimes get-'
`In tweed, that man!' barked Ridcully. `And no pointy hat!' Against a background of grumbling he turned then to the Librarian. `And as for you, sir ... a suit too. And a stovepipe hat. You need the height!'
'Ook!' said the Librarian.
`I am the Archchancellor, sir! I insist! And a false beard, I think. False eyebrows, too. Let Mr Darwin be your model here! These Victorians were very civilised people! Hair everywhere! Keep the knuckling to a minimum and they'll make you Prime Minister! Very well, gentlemen. Back here in half an hour!
The wizards assembled. A circle of white light appeared on the floor. They stepped inside, there was a change in the sounds made by Hex, and they vanished.
They landed knee-deep in the mire of a peat bog, causing bubbles
of foul air to burst around them.
`Mr Stibbons!' Ridcully bellowed.
`Sorry, sir, sorry,' said Ponder quickly. `Hex, raise us by two feet,
please.'
`Yes, but we're still soaked,' grumbled the Dean, as they floated
up in the air. `You seem to have, ah, "mucked up", Mr Stibbons!' 'No, sir, I wanted to show you a Charles Darwin in the wild,' said
Ponder. 'Here he comes now ...
A large and energetic young man bounded out of the weeds and went to clear a black pool with a vaulting pole. The pole immediately sank one-third of its length into the sucking ground and its athletic owner sailed off into the mud. He came up holding a small water plant. Oblivious of the noisome bubbling around him, he waved the plant triumphantly at some distant companions, pulled his pole out of the peat with some effort, and splashed away.
`Did he see us?' said Rincewind.
`No, not yet. That's young Darwin,' said Ponder. `Very keen on collecting all sorts of wildlife. Collecting was enormous popular among the English of this century. Bones, shells, butterflies, birds, other people's countries ... all sorts of thi
ngs.'
`Man after my own heart!' said Ridcully, cheerfully. `I had the best pressed lizard collection ever when I was that age!'
`Can't see a beagle anywhere, though,' said Rincewind, gloomily. He got edgy in the absence of his hat, and tried to stand under things.
The Chair of Indefinite Studies looked up from the thaumometer in his hand.
`No magic disturbance, no nothing,' he said, looking around at the
marshes. `Is Hex sure? The only strange thing here is us.'
`Let's get started, shall we?' said Ridcully. `Where to next?' `Hex, move us to London, will you?' said Ponder. `Location 7.' The wizards didn't apparently move, but the landscape around
them wavered and changed.
It became an alleyway. There were a lot of street noises nearby. `I'm sure you all read the briefing I prepared this morning.' said Ponder, brightly.
`Are you also sure were not back in Ankh-Morpork?' said Ridcully loudly. `I'd swear I can smell the river!'
`Ah, then perhaps I'd better just remind you of the important points,' said Ponder wearily. `The list of major things that might impede Darwin's progress-'
`I remember about the giant squid,' Rincewind volunteered.
`Hex can handle the giant squid,' said Ponder.
`Oh, shame. I was looking forward to that,' said Ridcully.
`No, sir,' said Ponder, as patiently as possible. `We have to deal with
people. Remember? We agreed last time it's not ethical to leave that
to Hex. Remember the rain of fat women?[1]
`That never actually happened,' said the Lecturer in Recent Runes,
wistfully.
`Quite so,' said Ridcully, firmly. `And just as well. Lead on, Mr Stibbons.'
`So much to do, so much to do,' muttered Ponder, leafing through
the paperwork. `I suppose we'd better do things in order ... so first,
we must see that Mr Habbakuk Souser's cook throws away the fish.'
It was a scullery boy who opened the back door, in a street of quite prosperous-looking houses. Ponder Stibbons raised his very tall hat.
`We wish to see - ' he consulted the clipboard ` - Mrs Boddy,' he said. `She is the cook here, I believe? Tell her we are the Committee for Public Sanitation, and the matter is urgent, so look sharp about it!'
`I hope you know what you're doing, Stibbons,' hissed Ridcully as the boy scurried 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.
[1] A rare meteorological phenomenon discussed briefly in The Science of Discworld II
`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.'
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 Godel, 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.
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