The metric of spacetime is not flat.
Instead, near a star, spacetime takes the form of a curved surface that bends to create a circular ‘valley’ in which the star sits. Light follows geodesies across the surface, and is ‘pulled down’ into the hole, because that path provides a short cut. Particles moving in spacetime at sublight speeds behave in the same way; they no longer follow straight lines, but are deflected towards the star, whence the Newtonian picture of a gravitational force.
Far from the star, this spacetime is very close indeed to Minkowski spacetime; that is, the gravitational effect falls off rapidly and soon becomes negligible. Spacetimes that look like Minkowski spacetime at large distances are said to be ‘asymptotically flat’. Remember that term: it’s important for making time machines. Most of our own universe is asymptotically flat, because massive bodies such as stars are scattered very thinly.
When setting up a spacetime, you can’t just bend things any way you like. The metric must obey the Einstein equations, which relate the motion of freely moving particles to the degree of distortion away from flat spacetime.
We’ve said a lot about how space and time behave, but what are they? To be honest, we haven’t a clue. The one thing we’re sure of is that appearances can be deceptive.
Tick.
Some physicists take that principle to extremes. Julian Barbour, in The End of Time, argues that from a quantum-mechanical point of view, time does not exist.
Ti —
In 1999, writing in New Scientist, he explained the idea roughly this way. At any instant, the state of every particle in the entire universe can be represented by a single point in a gigantic phase space, which he calls Platonia. Barbour and his colleague Bruno Bertotti found out how to make conventional physics work in Platonia. As time passes, the configuration of all particles in the universe is represented in Platonia as a moving point, so it traces out a path, just like a relativistic world-line. A Platonian deity could bring the points of that path into existence sequentially, and the particles would move, and time would seem to flow.
Quantum Platonia, however, is a much stranger place. Here, ‘quantum mechanics kills time’, as Barbour puts it. A quantum particle is not a point, but a fuzzy probability cloud. A quantum state of the universe is a fuzzy cloud in Platonia. The ‘size’ of that cloud, relative to that of Platonia itself, represents the probability that the universe is in one of the states that comprise the cloud. So we have to endow Platonia with a ‘probability mist’, whose density in any given region determines how probable it is for a cloud to occupy that region.
But, says Barbour, ‘there cannot be probabilities at different times, because Platonia itself is timeless. There can only be once-and-for-all probabilities for each possible configuration.’ There is only one probability mist, and it is always the same. In this set-up, time is an illusion. The future is not determined by the present – not because of the role of chance, but because there is no such thing as future or present.
By analogy, think of the childhood game of snakes and ladders. At each roll of the dice, players move their counters from square to square on a board; traditionally there are a hundred squares. Some are linked by ladders, and if you land at the bottom you immediately rise to the top; others are linked by snakes, and if you land at the top you immediately fall to the bottom. Whoever reaches the final square first wins.
To simplify the description, imagine someone playing solo snakes and ladders, so that there is only one counter on the board. Then at any instant, the ‘state’ of the game is determined by a single square: whichever one is currently occupied by the counter. In this analogy, the board itself becomes the phase space, our analogue of Platonia. The counter represents the entire universe. As the counter hops around, according to the rules of the game, the state of the ‘universe’ changes. The path that the counter follows – the list of squares that it successively occupies – is analogous to the world-line of the universe. In this interpretation, time does exist, because each successive move of the counter corresponds to one tick of the cosmic clock.
Quantum snakes and ladders is very different. The board is the same, but now all that matters is the probability with which the counter occupies any given square – not just at one stage of the game, but overall. For instance, the probability of being on the first square, at some stage in the game, is 1, because you always start there. The probability of being on the second square is 1/6, because the only way to get there is to throw a 1 with the dice on your first throw. And so on. Once we have calculated all these probabilities, we can forget about the rules of the game and the concept of a ‘move’. Now only the probabilities remain. This is the quantum version of the game, and it has no explicit moves, only probabilities. Since there are no moves, there is no notion of the ‘next’ move, and no sensible concept of time.
Our universe, Barbour tells us, is a quantum one, so it is like quantum snakes and ladders, and ‘time’ is a meaningless concept. So why do we naive humans imagine that time flows; that the universe (at least, the bit near us) passes through a linear sequence of changes?
To Barbour, the apparent flow of time is an illusion. He suggests that Platonian configurations which have high probability must contain within them ‘an appearance of history’. They will look as though they had a past. It’s a bit like the philosophers’ old chestnut: maybe the universe is being created anew every instant (as in Thief of Time), but at each moment, it is created along with apparent records of a lengthy past history. Such apparently historical clouds in Platonia are called time capsules. Now, among those high-probability configurations we find the arrangement of neurons in a conscious brain. In other words, the universe itself is timeless, but our brains are time capsules, high-probability configurations, and these automatically come along with the illusion that they have had a past history.
It’s a neat idea, if you like that sort of thing. But it hinges on Barbour’s claim that Platonia must be timeless because ‘there can only be once-and-for-all probabilities for each possible configuration’. This statement is remarkably reminiscent of one of Xeno’s – sorry, Zeno’s – paradoxes: the Arrow. Which, you recall, says that at each instant an arrow has a specific location, so it can’t be moving. Analogously, Barbour tells us that at each instant (if such a thing could exist) Platonia must have a specific probability mist, and deduces that this mist can’t change (so it doesn’t).
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 sequential-state 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.
1 Actually, foot.
2 Well, let’s not exaggerate. You can publish papers on it without risking losing your job. It’s certainly better than publishing nothing, which definitely will lose you your job.
3 Indeed, in the Back to the Future movie sequence, it was a car. A Delorean. Though it did need the assistance of a railway locomotive at one point.
4 Provided it is fired by someone who has been in the pub since lunchtime.
5 Actually this is the ambiguous puzuma, which travels at near-lightspeed (which on the Disc is about the speed of sound). If you see a puzuma, it’s not there. If you hear it, it’s not there either.
6 As one does. Palaeontologists have just announced that they have found remarkably well-preserved fossils in an East Anglian quarry, showing that giant hippos weighing six or seven tons – roughly twice the weight of modern hippos – wallowed in the rivers of Norfolk 600,000 years ago. It was a warm period sandwiched between two ice ages, probably a few degrees warmer than the present day (you can tell that from insect fossils) and hyenas prowled the banks in search of carrion.
7 See The Annotated Flatland by Edwin A. Abbott and Ian Stewart (Basic Books, 2002).
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 …
*
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 things.’
‘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 Studi
es 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 we’ve 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?’2
‘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 here we are the Committee for Public Sanitation, and the matter is urgent, so look sharp about it!’
Science of Discworld III Page 9