[1] Derived from a pun: m-bran for `membrane'. Opening up jokes about no-branes and p-branes. Oh well.
distinct topological types of surface: a sphere, a doughnut, two doughnuts joined together, three doughnuts ... and in more dimensions than two, there are more exotic possibilities.
Particles correspond to tiny closed strings that loop around the brane. There are lots of different ways to loop a string round a doughnut - once through the hole, twice, three times ... The physical laws depend on the shape of the brane and the paths followed by these loops.
The current favourite brane has six dimensions, making ten in all. The extra dimensions are thought to be curled up very tightly, smaller than the Planck length, which is the size at which the universe becomes grainy. It is virtually impossible to observe anything that small, because the graininess blurs everything and the fine detail cannot be seen. So there's no hope of observing any extra dimensions directly. However, there are several ways to infer their presence indirectly. In fact, the recently discovered acceleration in the rate of expansion of the universe can be explained in that manner. Of course, this explanation may not be correct: we need more evidence.
The ideas here change almost by the day, so we don't have to commit ourselves to the currently favoured six-dimensional set-up. We can contemplate any number of different branes and differently arranged loops. Each choice - call it a loopy brane - has a particular energy, related to the shape of the brane, how tightly it is curled up, and how tightly the loops wind round it. This energy is the 'vacuum energy' of the associated physical theory. In quantum mechanics, a vacuum is a seething mass of particles and antiparticles coming into existence for a brief instant before they collide and annihilate each other again. The vacuum energy measures how violently they seethe. We can use the vacuum energy to infer which loopy brane corresponds to our own universe, whose vacuum energy is extraordinarily small. Until recently it was thought to be zero, but it's now thought to be about 1/120plex units, where a unit is one Planck mass per cubic Planck length, which is a googol grammes per cubic metre.
We now encounter a cosmic `three bears' story. Macho Daddy Bear prefers a vacuum energy larger than +1/118plex units, but such a spacetime would be subject to local expansions far more energetic than a supernova. Wimpy Mummy Bear prefers a vacuum energy smaller than -1/120plex units (note the minus sign), but then spacetime contracts in a cosmic crunch and disappears. Baby Bear and Goldilocks like their vacuum energy to be `just right': somewhere in the incredibly tiny range between +1/118plex and -1/120plex units. That is the Goldilocks zone in which life as we know it might possibly exist.
It is no coincidence that we inhabit a universe whose vacuum energy lies in the Goldilocks zone, because we are life as we know it. If we lived in any other kind of universe, we would be life as we don't know it. Not impossible, but not us.
This is our old friend the anthropic principle, employed in an entirely sensible way to relate the way we function to the kind of universe that we need to function in. The deep question here is not `why do we live in a universe like that?', but `why does there exist a universe like that, for us to live in?' This is the vexed issue of cosmological fine-tuning, and the improbability of a random universe hitting just the right numbers is often used to prove that something - they always say `We don't know, could be an alien,' but what they're all thinking is: 'God'- must have set our universe up to be just right for us.
The string theorists are made of sterner stuff, and they have a more sensible answer.
In 2000 Bousso and Polchinski combined string theory with an earlier idea of Steven Weinberg to explain why we shouldn't be surprised that a universe with the right level of vacuum energy exists. Their basic idea is that the phase space of possible universes is absolutely gigantic. It is bigger than, say, 500plex. Those 500plex universes distribute their vacuum energies densely in the range -1 to +1 units. The resulting numbers are much more closely packed than the 1/118plex units that determine the scale of the `acceptable' range of vacuum energies for life as we know it. Although only a very tiny proportion of those 500plex universes fall inside that range, there are so many of them that that a tiny proportion is still absolutely gigantic - here, around 382plex. So a whacking great 382plex universes, from a phase space of 500plex loopy branes, are capable of supporting our kind of life.
However, that's still a very small proportion. If you pick a loopy brane at random, the odds are overwhelmingly great that it won't fall inside the Goldilocks range.
Not a problem. The string theorists have an answer to that. If you wait long enough, such a universe will necessarily come into being. In fact, all universes in the phase space of loopy branes will eventually become the `real' universe. And when the real universe's loopy brane gets into the Goldilocks range, the inhabitants of that universe will not know about all that waiting. Their sense of time will start from the instant when that particular loopy brane first occurred.
String theory not only tells us that we're here because we're here - it explains why a suitable `here' must exist.
The reason why all of those 500plex or so universes can legitimately be considered `real' in string theory stems from two features of that theory. The first is a systematic way to describe all the possible loopy branes that might occur. The second invokes a bit of quantum to explain why, in the long run, they will occur. Briefly: the phase space of loopy branes can be represented as an `energy landscape', which we'll name the branescape. Each position in the landscape corresponds to one possible choice of loopy brane; the height at that point corresponds to the associated vacuum energy.
Peaks of the branescape represent loopy branes with high vacuum energy, valleys represent loopy branes with low vacuum energy. Stable loopy branes lie in the valleys. Universes whose hidden dimensions look like those particular loopy branes are themselves stable ... so these are the ones that can exist, physically, for more than a split second.
In hilly districts of the world, the landscape is rugged, meaning that it has a lot of peaks and valleys. They get closer together than elsewhere, but they are still generally isolated from each other. The branescape is very rugged indeed, and it has a huge number of valleys. But all of the valleys' vacuum energies have to fit inside the range from -1 units to +1 units. With so many numbers to pack in, they get squashed very close together.
In order for a universe to support life as we know it, the vacuum energy has to lie in the Goldilocks zone where everything is just right. And there are so many loopy branes that a huge number of them must have vacuum energies that fall inside it.
Vastly more will fall outside that range, but never mind.
The theory has one major advantage: it explains why our universe has such a small vacuum energy, without requiring it to be zero - which, we now know, it isn't.
The upshot of all the maths, then, is that every stable universe sits in some valley of the branescape, and an awful lot of them (though a tiny proportion of the whole) lie in the Goldilocks range. But all of those universes are potential, not actual. There is only one real universe. So if we merely pick a loopy brane at random, the chance of hitting the Goldilocks zone is pretty much zero. You wouldn't bet on a horse at those odds, let alone a universe.
Fortunately, good old quantum gallops to our rescue. Quantum systems can, and do, `tunnel' from one energy valley to another. The uncertainty principle lets them borrow enough energy to do that, and then pay it back so quickly that the corresponding uncertainty about timing prevents anyone noticing. So, if you wait long enough - umptyplexplexplex years, perhaps, or umptyplexplexplexplex if that's too short - then a single quantum universe will explore every valley in the entire branescape. Along the way, at some stage it finds itself in a Goldilocks valley. Life like ours then arises, and wonders why it's there.
It's not aware of the umptyplexplexplexplex years that have already passed in the multiverse: just of the few billion that have passed since the wandering universe tunnelled its way into the Goldi
locks range. Now, and only now, do its human-like inhabitants start to ask why it's possible for them to exist, given such ridiculous odds to the contrary. Eventually, if they're bright enough, they work out that thanks to the branescape and quantum, the true odds are a dead certainty.
It's a beautiful story, even if it turns out to be wrong.
FIFTEEN
AUDITORS OF REALITY
IT WAS ONE HOUR LATER. Wizards were ranged in rows
across the width of the Great Hall in a variety of costumes, but mostly in what might be called Early Trouser; despite Rincewind's view on nudity, a grubby shirt and pants would pass without comment in many ages and countries and lead to fewer arrests.
'Right, then,' said Ridcully, striding along the ranks 'We've kept all this very simple so that even professors can understand! Ponder Stibbons has given all of you your tasks!' He stopped in front of a middle-aged wizard. 'You, sir, who are you?'
'Don't you know, sir?' said the wizard, taken aback.
'Slipped m' mind, man!' said Ridcully. 'Big university, can't be expected to recognise everyone!'
'It's Pennysmart, sir. Professor of Extreme Horticulture.' 'Any good at it?'
'Yes, sir!'
'Any students?'
'No, sir!' said Pennysmart, looking offended.
'That's what I like to hear! And what will you be doin' today?' 'First, it appears, I shall be dropped waist-deep in a lagoon in the, the -' he stopped, and fumbled a piece of paper out of his pocket
'- Keeling islands, where I shall attack the sand bottom round me with this rake,' he held up the implement, 'and then return here as soon as I see any humans.'
'And how will you do that?'
'Say aloud, "Return Me, Hex",' said Pennysmart, smartly.
'Well done, good man,' said the Archchancellor. He raised his voice. 'Remember that, everyone! Exactly those words! Write them down if you can't remember them. Hex will bring you back on the lawn out side the building. There will be hundreds of you and many of you have several tasks, so we don't want any collisions! Now, if-'
'Excuse me,' said Pennysmart, raising a hand. 'Yes?'
'Why will I be standing in a lagoon flailing around with a rake, please?'
'Because if you don't do that, Darwin will tread on the dorsal spine
of an extremely poisonous fish,' said Ponder Stibbons. `Now-' 'Excuse me again, please,' Pennysmart said. 'Yes?'
'Why won't I tread on this fish?'
'Because you will be lookin' where you are treadin', Mr Pennysmart,' roared Ridcully.
But a forest of other hands had gone up. About the only wizard without a hand aloft was Rincewind, who was staring gloomily at his feet.
'What's all this about?' said the Archchancellor, irritably. 'Why .have I got to move a chair six inches?'
'Why have I got to fill up a hole in the middle of a prairie?' 'Why have I got to bide a pair of trousers?' 'Why have I got to stuff a letter box full of starved snails?' Ponder waved his clipboard wildly to silence the clamour. 'Because otherwise Darwin would have fallen off a chair or been thrown from a horse or would have been struck by a stone hurled
by a rioter or an unwise letter would have reached its destination,' he said. 'But there are more than two thousand tasks, so I can't explain every one. Some of them are the start of a quite astonishing causal chain.'
`We are supposed to develop questioning minds, you know,' someone muttered.
`Yes, but not regarding university policy!' said Ridcully. `You all have very simple jobs to do! Gentlemen, Mr Stibbons will call out your names, and you will step smartly into the circle! Over to you, Mr Stibbons!'
Ponder Stibbons picked up a different clipboard. He was beginning to collect clipboards. They proclaimed order in an increasingly hard-to-understand world. That's all I've ever really wanted, he thought. I just want to feel that things are being ticked off properly.
`Now, chaps,' he said. `This should not be hard, as the Archchancellor has said. If at all possible don't talk to anyone and don't touch anything. In and out, that's the ticket. I want this done fast. I have a ... theory about that. So don't waste time, wherever you go. Are we all ready? Very well ... Aardvarker, Professor A ... '
One by one, with confidence or trepidation or a mixture of both, wizards stepped into Hex's circle of light and vanished. As they did so, little pointy-hatted wizard symbols appeared at points in the tangle of lights above.
Rincewind watched gloomily, and didn't join in the ragged cheer as, one by one, red circles began to wink out.
Ponder had taken him aside earlier and had explained that, since Rincewind was so experienced at this sort of thing, he was going to be given four of the most, er, interesting tasks. That was how he had put it: 'er, interesting'. Rincewind knew all about 'er, interesting'. There was a giant squid out there with his name on it, that's what it meant.
A movement at the end on the hall made him look around. It was a chest, a metal-bound box of the kind favoured by people who bury treasure, and it walked on hundreds of little pink legs. He groaned. He'd left it asleep on the wardrobe in his bedroom, with its feet in the air.
'Hmm?' he said.
`Rincewind! Off you go, best of luck!' Ponder repeated. `Hurry up!' There was nothing for it. Rincewind walked into the circle, and fell over as the ship moved gently under him.
It was dawn, and a clammy sea mist was drifting across the deck. Rigging creaked, the water lapped far below. There was no other sound. The air smelled warm and exotic.
There was a small cannon only a few feet away. Rincewind knew about them. He was the only wizard to have seen one, over in the Agatean Empire, where they were known as `Barking Dogs'. He was sure that one of the rules associated with them was `do not stand in front'.
Slowly, he reached inside his shirt and pulled out his pointy hat. It was red, or rather, it was the colour that red becomes after being washed, eaten, retrieved, scorched, buried, crushed, engulfed, washed again and wrung out far, far too often.
No wearing of pointy hats? Were they mad? He pulled at it a bit to get it back to its comfortable shapeless shape, and put it on. That felt much better. A pointy hat meant you weren't just anyone.
He unrolled his instructions.
1. Remove ball from `cannon'
There was no one around. There was a stack of metal balls by the cannon. Rincewind pulled the barrel around with some effort, felt down the hole, and grunted as his fingers touched the top of another ball at the far end.
How could he get it out? The way to get a ball out of a Barking Dog was to set a match to its tail, but Ponder had said this wasn't an option. He cast around, and saw a bundle of tools by the stack; one was a rod with an end like a super-corkscrew.
Carefully, he pushed it down the cannon, wincing at every clink. Twice he felt the curved springy bits engage with the ball, and twice it came away and rolled back with a thud.
At the third attempt he was able to get the tapped ball almost out of the mouth of the barrel, and slid his fingers under it.
Well, that wasn't too hard, was it? He dropped it over the side, where the sea swallowed it with a 'plomp!'
This caused no stir anywhere. Job done, and nothing horrible had happened at all! He pulled a scrap of paper out of his pocket. It was important to get the words right.
`Return-' he began, and stopped. With a little metallic grinding noise, another ball rolled gently off the pile, across the deck, and leapt into the cannon's mouth.
`O-kay,' said Rincewind slowly. Of course. Obviously. Why had he thought otherwise for even one second?
Sighing, he picked up the ball grasper, rammed it down the barrel, caught the ball, and jerked it out so hard that it would have made a giveaway noise hitting the deck. Fortunately, it landed on Rincewind's foot.
A little metallic sound disturbed him while he was lying across the barrel making the traditional 'gheeee' noise of those who are screaming through clenched teeth.
It was the noise of another cannon ball
rolling across the deck. He jumped on it, picked it up, and felt a slight resistance trying to tug it out of his hands. He wrenched against the invisible force, spun around and the ball flew out of his hands and over the rail.
This time the 'plomp!' caused an interrogatory mumble from below decks.
The last remaining ball started to roll towards the cannon.
`Oh no you don't!' snarled Rincewind, and grabbed it. Again the force tried to pull the ball away from him, but he clung on tightly.
There was the sound of footsteps climbing stairs. Somewhere close, in the fog, someone sounded angry.
Then in the billows in front of Rincewind there was ... something. He couldn't make out the shape, but it disturbed the fog, making an outline of sorts. It looked like
It let go as someone hurried closer. Rincewind growled in triumph, staggered backwards, tipped over the rail and, still clutching the cannon ball, went `plomp!'
`Look at the red circles, sir!' shouted Ponder.
Erratically, in the drifting tangle of lights, the red circles were winking out. The yellow line was extending.
`That's the style, Mr Stibbons!' the Archchancellor roared. `Keep pounding away!'
Wizards were scuttling through the hall, getting fresh instructions, catching their breath and disappearing in the circle again.
Ridcully nodded at the stretcher containing the screaming Pennysmart, as it was hurried away to the Infirmary.
`Never seen that shade of purple on a leg,' he said. `I told him to look where he was going. You heard me, didn't you?'
`He says he was dropped right on top of the fish,' said Ponder. `I'm afraid Hex is running at the very limit of his power, sir. We're bending an entire timeline. You've got to expect some accidents. A few of the returning wizards are reappearing in the fountain. We just have to accept that it's better than them reappearing inside walls.'
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