The Science of Discworld III - Darwin's Watch

6. BORROWED TIME

THE EVER-BRANCHING LEGS OF the Trousers of Time are a metaphor (unless you are a quantum physicist, in which case they represent a certain mathematical view of reality) for the many paths that history might have taken if events had been slightly different. Later, we'll think about all those legs, but for now, we restrict attention to one trouser. One timeline. What exactly is time?

We know what it is on Discworld. `Time', states The New Discworld Companion, `is one of the Discworld's most secretive anthropomorphic personifications. It is hazarded that time is female (she waits for no man) but she has never been seen in the mundane worlds, having always gone somewhere else just a moment before. In her chronophonic castle, made up of endless glass rooms, she does at, er, times, materialise into a tall woman with dark hair, wearing a long red-and-black dress.'

Tick.

Even Discworld has trouble with time. In Roundworld it's worse. There was a time (there we go) when space and time were considered to be totally different things. Space had, or was, extension - it sort of spread itself around, and you could move through it at will. Within reason, maybe 20 miles (30km) a day on a good horse if the tracks weren't too muddy and the highwaymen weren't too obtrusive.

Tick.

Time, in contrast, moved of its own volition and took you along with it. Time just passed, at a fixed speed of one hour per hour, always in the direction of the future. The past had already happened, the present was happening right now - oops, gone already - and the future had yet to happen, but by jingo, it would, you mark my words, when it was good and ready.

Tick.

You could choose where you went in space, but you couldn't choose when you went in time. You couldn't visit the past to find out what had really happened, or visit the future to find out what fate had in store for you; you just had to wait and find out. So time was completely different from space. Space was three-dimensional, with three independent directions: left/right, back/forward, up/down. Time just was.

Tick.

Then along came Einstein, and time started to get mixed up with space. Time-like directions were still different from space-like ones, in some ways, but you could mix them up a bit. You could borrow time here and pay it back somewhere else. Even so, you couldn't head off into the future and find yourself back in your own past. That would be time travel, which played no part in physics.

What science abhors, the arts crave. Time travel may be a physical impossibility, but it is a wonderful narrative device for writers, because it allows the story to move to past, present, or future, at will. Of course you don't need a time machine to do that - the flashback is a standard literary device. But it's fun (and respectful to narrativium) to have some kind of rationale that fits into the story itself. Victorian writers liked to use dreams; a good example is Charles Dickens's A Christmas Carol of 1843, with its ghosts of Christmas past, present, and yet-to-come. There is even a literary subgenre of 'timeslip romances', some of them really quite steamy. The French ones.

Time travel causes problems if you treat it as more than just a literary device. When allied to free will, it leads to paradoxes. The ultimate cliche here is the `grandfather paradox', which goes back to Rene Barjavel's story Le Voyageur Imprudent. You go back in time and kill your grandfather, but because your father is then not born, neither are you, so you can't go back to kill him ... Quite why it's always your grandfather isn't clear (except as a sign that it's a cliche, a low-bred form of narrativium). Killing your father or mother would have the same paradoxical consequences. And so might the slaughter of a Cretaceous butterfly, as in Ray Bradbury's 1952 short story `A Sound of Thunder', in which a butterfly's accidental demise at the hands[14] of an unwitting time traveller changes present-day politics for the worse.

Another celebrated time paradox is the cumulative audience paradox. Certain events, the standard one being the Crucifixion, are so endowed with narrativium that any self-respecting time tourist will insist on seeing them. The inevitable consequence is that anyone who visits the Crucifixion will find Christ surrounded by thousands, if not millions, of time travellers. A third is the perpetual investment paradox. Put your money in a bank account in 1955, take it out in 2005, with accumulated interest, then take it back to 1955 and put it in again ... Be careful to use something like gold, not notes - notes from 2005 won't be valid in 1955. Robert Silverberg's Up the Line is about the Time Service, a force of time police whose job is to prevent such paradoxes from getting out of hand. A similar theme occurs in Isaac Asimov's The End of Eternity.

An entire class of paradoxes arises from time loops, closed loops of causality in which events only get started because someone comes from the future to initiate them. For example, the easiest way for today's humanity to get hold of a time machine is if someone is presented with one by a time traveller from the far future, when such machines have already been invented. He or she then reverse engineers the machine to find out how it works, and these principles later form the basis for the future invention of the machine. Two classic stories of this type are Robert Heinlein's 'By His Bootstraps' and `All you Zombies', the second being noteworthy for a protagonist who becomes his own father and his own mother (via a sex change). David Gerrold took this idea to extremes in The Man Who Folded Himself.

Science-fiction authors are divided on whether time paradoxes always neatly unwrap themselves to produce consistent results, or whether it is genuinely possible, in their fictional setting, to change the past or the present. (No one worries much about changing the future, mind you, presumably because `free will' amounts to precisely that. We all change the future, from what it might have been to what it actually becomes, thousands of times every day. Or so we fondly imagine.) So some authors write of attempts to kill your grandfather that, by some neat twist, bring you into existence anyway. For example, your true father was not his son at all, but a man he killed. By mistakenly eliminating the wrong grandfather, you ensure that your true father survives to sire you. Others, like Asimov and Silverberg, set up entire organisations dedicated to making sure that the past, hence the present, remains intact. Which may or may not work.

The paradoxes associated with time travel are part of the subject's fascination, but they do rather point towards the conclusion that time travel is a logical impossibility, let alone a physical one. So we are happy to allow the wizards of Unseen University, whose world runs on magic, the facility to wander at will up and down the Roundworld timeline, switching history from one parallel universe to another, trying to get Charles Darwin - or somebody - to write That Book. The wizards live in Discworld, they operate outside Roundworld constraints. But we don't really imagine that Roundworld people could do the same, without external assistance, using only Roundworld science.

Strangely, many scientists at the frontiers of today's physics don't agree. To them, time travel has become an entirely respectable[15] research topic, paradoxes notwithstanding. It seems that there is nothing in the `laws' of physics, as we currently understand them, that forbids time travel. The paradoxes are apparent rather than real; they can be `resolved' without violating physical law, as we will see in Chapter 8. That may be a flaw in today's physics, as Stephen Hawking maintains; his `chronology protection conjecture' states that as yet unknown physical laws conspire to shut down any time machine just before it gets assembled - a built-in cosmological time cop.

On the other hand, the possibility of time travel may be a profound statement about the universe. We probably won't know for sure until we get to tackle the issue using tomorrow's physics. And it's worth remarking that we don't really understand time, let alone how to travel through it.

Although (apparently) the laws of physics do not forbid time travel, it turns out that they do make it very difficult. One theoretical scheme for achieving that goal, which involves towing black holes around very fast, requires rather more energy than is contained in the entire universe. This is a bit of a bummer, and it does seem to rule out the typical science fiction time machine, about the size of a car[16].

The most extensive descriptions of Discworld time are found in Thief of Time. The ingredients for this novel include a member of the Guild of Clockmakers, Jeremy Clockson, who is determined to make a completely accurate clock. However, he is up against a theoretical barrier, the paradoxes of the Ephebian philosopher Xeno, which are first mentioned in Pyramids. A Roundworld philosopher with an oddly similar name, Zeno of Elea, born around 490 BC, stated four paradoxes about the relation between space, time and motion. He is Xeno's Roundworld counterpart, and his paradoxes bear a curious resemblance to the Ephebian philosopher's. Xeno proved by logic alone that an arrow cannot hit a running man[17], and that the tortoise is the fastest animal on the Disc[18]. He combined both in one experiment, by shooting an arrow at a tortoise that was racing against a hare. The arrow hit the hare by mistake, and the tortoise won, which proved that he was right. In Pyramids, Xeno describes the thinking behind this experiment.

"s quite simple,' said Xeno. `Look, let's say this olive stone is an arrow and this, and this -' he cast around aimlessly -'and this stunned seagull is the tortoise, right? Now, when you fire the arrow it goes from here to the seag- the tortoise, am I right?'

`I suppose so, but-'

`But, by this time, the seagu- the tortoise has moved on a bit, hasn't he? Am I right?'

'I suppose so,' said Teppic, helplessly. Xeno gave him a look of triumph.

`So the arrow has to go a bit further, doesn't it, to where the tortoise is now. Meanwhile the tortoise has flow- moved on, not much, I'll grant you, but it doesn't have to be much. Am I right? So the arrow has a bit further to go, but the point is that by the time it gets to where the tortoise is now the tortoise isn't there. So if the tortoise keeps moving, the arrow will never hit it. It'll keep getting closer and closer, but it'll never hit it. QED.'

Zeno has a similar set-up, though he garbles it into two paradoxes. The first, called the Dichotomy, states that motion is impossible, because before you can get anywhere, you have to get halfway, and before you can get there, you have to get halfway to that, and so on for ever ... so you have to do infinitely many things to get started, which is silly. The second, Achilles and the Tortoise, is pretty much the paradox enunciated by Xeno, but with the hare replaced by the Greek hero Achilles. Achilles runs faster than the tortoise - face it, anyone can run faster than a tortoise - but he starts a bit behind, and can never catch up because whenever he reaches the place where the tortoise was, it's moved on a bit. Like the ambiguous puzuma, by the time you get to it, it's not there. The third paradox says that a moving arrow isn't moving. Time must be divided into successive instants, and at each instant the arrow occupies a definite position, so it must be at rest. If it's always at rest, it can't move. The fourth of Zeno's paradoxes, the Moving Rows (or Stadium), is more technical to describe, but it boils down to this. Suppose three bodies are level with each other, and in the smallest instant of time one moves the smallest possible distance to the right, while the other moves the smallest possible distance to the left. Then those two bodies have moved apart by twice the smallest distance, taking the smallest instant of time to do that. So when they were just the smallest distance apart, halfway to their final destinations, time must have changed by half the smallest possible instant of time. Which would be smaller, which is crazy.

There is a serious intent to Zeno's paradoxes, and a reason why there are four of them. The Greek philosophers of Roundworld antiquity were arguing whether space and time were discrete, made up of indivisible tiny units, or continuous - infinitely divisible. Zeno's four paradoxes neatly dispose of all four combinations of continuous/discrete for space with continuous/discrete for time, neatly stuffing everyone else's theories, which is how you make your mark in philosophical circles. For instance, the Moving Rows paradox shows that having both space and time discrete is contradictory.

Zeno's paradoxes still show up today in some areas of theoretical physics and mathematics, although Achilles and the Tortoise can be dealt with by agreeing that if space and time are both continuous, then infinitely many things can (indeed must) happen in a finite time. The Arrow paradox can be resolved by noting that in the general mathematical treatment of classical mechanics, known as Hamiltonian mechanics after the great (and drunken) Irish mathematician Sir William Rowan Hamilton, the state of a body is given by two quantities, not one. As well as position it also has momentum, a disguised version of velocity. The two are related by the body's motion, but they are conceptually distinct. All you see is position; momentum is observable only through its effect on the subsequent positions. A body in a given position with zero momentum is not moving at that instant, and so will not go anywhere, whereas one in the same position with non-zero momentum - which appears identical - is moving, even though instantaneously it stays in the same place.

Got that?

Anyway, we were talking about Thief of Time, and thanks to Xeno we've not yet got past page 21. The main point is that Discworld time is malleable, so the laws of narrative imperative sometimes need a little help to make sure that the narrative does what the imperative says it should.

Tick.

Lady Myria Lejean is an Auditor of reality, who has temporarily assumed human form. Discworld is relentlessly animistic; virtually everything is conscious on some level, including basic physics. The Auditors police the laws of nature; they would very likely fine you for exceeding the speed of light. They normally take the form of small grey robes with a cowl - and nothing inside. They are the ultimate bureaucrats. Lejean points out to Jeremy that the perfect clock must be able to measure Xeno's smallest unit of time. `It must exist, mustn't it? Consider the present. It must have a length, because one end of it is connected to the past and the other is connected to the future, and if it didn't have a length then the present. couldn't exist at all. There would be no time for it to be the present in.'

Her views correspond rather closely to current theories of the psychology of the perception of time. Our brains perceive an `instant' as an extended, though brief, period of time. This is analogous to the way discrete rods and cones in the retina seem to perceive individual points, but actually sample a small region of space. The brain accepts coarse-grained inputs and smooths them out.

Lejean is explaining Xeno to Jeremy because she has a hidden agenda: if Jeremy succeeds in making the perfect clock, then time will stop. This will make the Auditors' task as clerks of the universe much simpler, because humans are always moving things around, which makes it difficult to keep track of their locations in time and space.

Tick.

Near the Discworld Hub, in a high, green valley, lies the monastery of Oi Dong, where live the fighting monks of the order of Wen, otherwise known as History Monks. They have taken upon themselves the task of ensuring that the right history happens in the right order. The monks know what is right because they guard the History Books, which are not records of what did happen, but instructions for what should.

A youngster named Ludd, a foundling brought up by the Thieves' Guild, where he was an exceptionally talented student, has been recruited to the ranks of the History Monks and given the name Lobsang. The monks' main technological aids are procrastinators, huge spinning machines that store and move time. With a procrastinator, you can borrow time and pay it back later. Lobsang wouldn't dream of living on borrowed time, though - but if it wasn't nailed down, he would almost certainly steal it. He can steal anything, and usually does. And, thanks to the procrastinators, time is not nailed down.

If you haven't got the joke by now, take another look at the title.

Lejean's plan works; Jeremy builds his clock.

Time stops, which is what the Auditors wanted. Not only on Discworld: temporal stasis expands across the universe at the speed of light. Soon, everything will stop. The History Monks are powerless, for they, too, have stopped. Only Susan Sto Helit, Death's granddaughter, can get time started again. And Ronnie Soak, who used to be Kaos, the Fifth Horseman of the Apocralypse, but left because of artistic disputes before they became famous ... Fortunately, the Auditors like obeying rules, and DO NOT FEED THE ELEPHANT really perplexes them when there is no elephant to feed. Fatally, they also have a love-hate relationship with chocolate. They are living on stolen time.

A procrastinator is a sort of time machine, but it moves time itself, instead of moving people through time. Moreover, it's fact, not fiction, as is all of Discworld to those who live there. On Roundworld, the first fictional time machine, as opposed to dreams or narrative timeslip, seems to have been invented by Edward Mitchell, an editor for the New York Sun newspaper. In 1881 he published an anonymous story, `The Clock That Went Backward', in his paper. The most celebrated time-travel gadget appears in Herbert George Wells's novel The Time Machine of 1895, and this set a standard for all that followed. The novel tells of a Victorian inventor who builds a time machine and travels into the far future. There he finds that humanity has speciated into two distinct types - the nasty Morlocks, who live deep inside caverns, and the ethereal Eloi, who are preyed on by the Morlocks and are too indolent to do anything about it. Several movies, all fairly ghastly, have been based on the book.

The novel had inauspicious beginnings. Wells studied biology, mathematics, physics, geology, drawing, and astrophysics at the Normal School of Science, which became the Royal College of Science and eventually merged with Imperial College of Science and Technology. While a student there, he began the work that led up to The Time Machine. His first time-travel story `The Chronic Argonauts' appeared in 1888 in the Science Schools Journal, which Wells helped to found. The protagonist voyages into the past and commits a murder. The story offers no rationale for time travel and is more of a mad-scientist tale in the tradition of Mary Shelley's Frankenstein, but nowhere near as well written. Wells later destroyed every copy of it he could locate, because it embarrassed him so much. It lacked even the paradoxical element of the 1891 Tourmalin's Time Cheques by Thomas Anstey Guthrie, which introduced many of the standard time-travel paradoxes.

Over the following three years, Wells produced two more versions of his time-travel story, now lost, but along the way the storyline mutated into a far-future vision of the human race. The next version appeared in 1894 in the National Observer magazine, as three connected tales with the title `The Time Machine'. This version has many features in common with the final novel, but before publication was complete, the editor of the magazine moved to the New Review. There he commissioned the same series again, but this time Wells made substantial changes. The manuscripts include many scenes that were never printed: the hero journeys into the past, running into a prehistoric hippopotamus[19] and meeting the Puritans in 1645. The published magazine version is very similar to the one that appeared in book form in 1895. In this version the Time Traveller moves only into the future, where he finds out what will happen to the human race, which splits into the languid Eloi and the horrid Morlocks - both equally distasteful.

Where did Wells get the idea? The standard SF writer's reply to this question is that `you make it up', but we have some fairly specific information in this case. In a foreword to the 1932 edition, Wells says that he was motivated by `student discussions in the laboratories and debating society of the Royal College of Science in the eighties'. According to Wells's son, the idea came from a paper on the fourth dimension read by another student. In the introduction to the novel, the Time Traveller (he is never named, but in the early version he is Dr Nebo-gipfel, so perhaps it's just as well) invokes the fourth dimension to explain why such a machine is possible:

`But wait a moment. Can an instantaneous cube exist?' `Don't follow you,' said Filby.

`Can a cube that does not last for any time at all, have a real existence?'

Filby became pensive.

`Clearly,' the Time Traveller proceeded, `any real body must have extension in four directions: it must have Length, Breadth, Thickness, and - Duration ...

.. There are really four dimensions, three which we call the three planes of Space, and a fourth, Time. There is, however, a tendency to draw an unreal distinction between the former three dimensions and the latter, because it happens that our consciousness moves intermittently in one direction along the latter from the beginning to the end of our lives ...

.. But some philosophical people have been asking why three dimensions particularly - why not another direction at right angles to the three? - and have even tried to construct a Four-Dimensional geometry. Professor Simon Newcomb was expounding this to the New York Mathematical Society only a month or so ago.'

The notion of time as a fourth dimension was becoming common scientific currency in the late Victorian era. The mathematicians had started it, by wondering what a dimension was, and deciding that it need not be a direction in real space. A dimension was just a quantity that could be varied, and the number of dimensions was the largest number of such quantities that could all be varied independently. Thus the Discworld thaum, the basic particle of magic, is actually composed of resons, which come in at least five flavours: up, down, sideways, sex appeal, and peppermint. The thaum is therefore at least five-dimensional, assuming that up and down are independent, which is likely because it's quantum.

In the 1700s the foundling mathematician Jean le Rond D'Alembert (his middle name is that of the church where he was abandoned as a baby) suggested thinking of time as a fourth dimension in an article in the Reasoned Encyclopaedia or Dictionary of Sciences, Arts, and Crafts. Another mathematician, Joseph-Louis Lagrange, used time as a fourth dimension in his Analytical Mechanics of 1788, and his Theory of Analytic Functions of 1797 explicitly states: `We may regard mechanics as a geometry of four dimensions.'

It took a while for the idea to sink in, but by Victorian times mathematicians were routinely combining space and time into a single entity. They didn't (yet) call it spacetime, but they could see that it had four dimensions: three of space plus one of time. Journalists and the lay public soon began to refer to time as the fourth dimension, because they couldn't think of another one, and to talk as if scientists had been looking for it for ages and had just found it. Newcomb wrote about the science of four-dimensional space from 1877, and spoke about it to the New York Mathematical Society in 1893.

Wells's mention of Newcomb suggests a link to one of the more colourful members of Victorian society, the writer Charles Howard Hinton. Hinton's primary claim to fame is his enthusiastic promotion of `the' fourth dimension. He was a talented mathematician with a genuine flair for four-dimensional geometry, and in 1880 he published `What is the Fourth Dimension?' in the Dublin University Magazine, which was reprinted in the Cheltenham Ladies' Gazette a year later. In 1884 it reappeared as a pamphlet with the subtitle `Ghosts Explained'. Hinton, something of a mystic, related the fourth dimension to pseudoscientific topics ranging from ghosts to the afterlife. A ghost can easily appear from, and disappear along, a fourth dimension, for instance, just as a coin can appear on, and disappear from, a tabletop, by moving along `the' third dimension.

Charles Hinton was influenced by the unorthodox views of his surgeon father James, a collaborator of Havelock Ellis, who outraged Victorian society with his studies of human sexual behaviour. Hinton the elder advocated free love and polygamy, and eventually headed a cult. Hinton the younger also had an eventful private life: in 1886 he fled to Japan, having been convicted of bigamy at the Old Bailey. In 1893 he left Japan to become a mathematics instructor at Princeton University, where he invented a baseball-pitching machine that used gunpowder to propel the balls, like a cannon. After several accidents the device was abandoned and Hinton lost his job, but his continuing efforts to promote the fourth dimension were more successful. He wrote about it in popular magazines like Harper 's Weekly, McClure's, and Science. He died suddenly of a cerebral haemorrhage in 1907, at the annual dinner of the Society of Philanthropic Enquiry, having just completed a toast to female philosophers.

It was probably Hinton who put Wells on to the narrative possibilities of time as the fourth dimension. The evidence is indirect but compelling. Newcomb definitely knew Hinton: he once got Hinton a job. We don't know whether Wells ever met Hinton, but there is circumstantial evidence of a close connection. For example, the term `scientific romance' was coined by Hinton in titles of his collected speculative essays in 1884 and 1886, and Wells later used the same phrase to describe his own stories. Moreover, Wells was a regular reader of Nature, which reviewed Hinton's first series of Scientific Romances (favourably) in 1885 and summarised some of his ideas on the fourth dimension.

In all likelihood, Hinton was also partially responsible for another Victorian transdimensional saga, Edwin A. Abbott's Flatland of 1884. The tale is about A. Square, who lives in the Euclidean plane, a twodimensional society of triangles, hexagons and circles, and doesn't believe in the third dimension until a passing sphere drops him in it. By analogy, Victorians who didn't believe in the fourth dimension were equally blinkered. A subtext is a satire on Victorian treatment of women and the poor. Many of Abbott's ingredients closely resemble elements found in Hinton's stories[20].

Most of the physics of time travel is general relativity, with a dash of quantum mechanics. As far as the wizards of Unseen University are concerned, all this stuff is `quantum' - a universal intellectual getout-of-jail card - so you can use it to explain virtually anything, however bizarre. Indeed, the more bizarre, the better. You're about to get a solid dose of quantum in Chapter 8. Here we'll set things up by providing a quick primer on Einstein's theories of relativity: special and general.

As we explained in The Science of Discworld, `relativity' is a silly name. It should have been 'absolutivity'. The whole point of special relativity is not that `everything is relative', but that one thing - the speed of light - is unexpectedly absolute. Shine a torch from a moving car, says Einstein: the extra speed of the car will have no effect on the speed of the light. This contrasts dramatically with old-fashioned Newtonian physics, where the light from a moving torch would go faster, acquiring the speed of the car in addition to its own inherent speed. If you throw a ball from a moving car, that's what happens. If you throw light, it should do the same, but it doesn't. Despite the shock to human intuition, experiments show that Roundworld really does behave relativistically. We don't notice because the difference between Newtonian and Einsteinian physics becomes noticeable only when speeds get close to that of light.

Special relativity was inevitable; scientists were bound to think of it. Its seeds were already sown in 1873 when James Clerk Maxwell wrote down his equations for electromagnetism. Those equations make sense in a `moving frame' - when observations are made by a moving observer - only if the speed of light is absolute. Several mathematicians, among them Henri Poincare and Hermann Minkowski, realised this and anticipated Einstein on a mathematical level, but it was Einstein who first took the ideas seriously as physics. As he pointed out in 1905, the physical consequences are bizarre. Objects shrink as they approach the speed of light, time slows to a crawl, and mass becomes infinite. Nothing (well, no thing) can travel faster than light, and mass can turn into energy.

In 1908 Minkowski found a simple way to visualise relativistic physics, now called Minkowski spacetime. In Newtonian physics, space has three fixed coordinates - left/right, front/back, up/down. Space and time were thought to be independent. But in the relativistic setting, Minkowski treated time as an extra coordinate in its own right. A fourth coordinate, a fourth independent direction ... a fourth dimension. Three-dimensional space became four-dimensional spacetime. But Minkowski's treatment of time added a new twist to the old idea of D'Alembert and Lagrange. Time could, to some extent, be swapped with space. Time, like space, became geometrical.

We can see this in the relativistic treatment of a moving particle. In Newtonian physics, the particle sits in space, and as time passes, it moves around. Newtonian physics views a moving particle the way we view a movie. Relativity, though, views a moving particle as the sequence of still frames that make up that movie. This lends relativity an explicit air of determinism. The movie frames already exist before you run the movie. Past, present and future are already there. As time flows, and the movie runs, we discover what fate has in store for us - but fate is really destiny, inevitable, inescapable. Yes - the movie frames could perhaps come into existence one by one, with the newest one being the present, but it's not possible to do this consistently for every observer.

Relativistic spacetime = geometric narrativium.

Geometrically, a moving point traces out a curve. Think of the particle as the point of a pencil, and spacetime as a sheet of paper, with space running horizontally and time vertically. As the pencil moves, it leaves a line behind on the paper. So, as a particle moves, it traces out a curve in spacetime called its world-line. If the particle moves at a constant speed, the world-line is straight. Particles that move very slowly cover a small amount of space in a lot of time, so their world-lines are close to the vertical; particles that move very fast cover a lot of space in very little time, so their world-lines are nearly horizontal. In between, running diagonally, are the world-lines of particles that cover a given amount of space in the same amount of time - measured in the right units. Those units are chosen to correspond via the speed of light - say years for time and light-years for space. What covers one light-year of space in one year of time? Light, of course. So diagonal world-lines correspond to particles of light - photons - or anything else that can move at the same speed.

Relativity forbids bodies that move faster than light. The worldlines that correspond to such bodies are called timelike curves, and the timelike curves passing through a given event form a cone, called its `light cone'. Actually, this is like two cones stuck together at their sharp tips, one pointing forward, the other backward. The forwardpointing cone contains the future of the event, all the points in spacetime that it could possibly influence. The backward-pointing cone contains its past, the events that could possibly influence it. Everything else is forbidden territory, elsewheres and elsewhens that have no possible causal connections to the chosen event.

Minkowski spacetime is said to be `flat' - it represents the motion of particles when no forces are acting on them. Forces change the motion, and the most important force is gravity. Einstein invented general relativity in order to incorporate gravity into special relativity. In Newtonian physics, gravity is a force: it pulls particles away from the straight lines that they would naturally follow if no force were acting. In general relativity, gravity is a geometric feature of the universe - a form of spacetime curvature.

In Minkowski spacetime, points represent events, which have a location in both space and time. The `distance' between two events must capture how far apart they are in space, and how far apart they are in time. It turns out that the way to do this is, roughly speaking, to take the distance between them in space and subtract the distance between them in time. This quantity is called the interval between the two events. If, instead, you did what seems obvious and added the time-distance to the space-distance, then space and time would be on exactly the same physical footing. However, there are clear differences: free motion in space is easy, but free motion in time is not. Subtracting the time-difference reflects this distinction; mathematically it amounts to considering time as imaginary space - space multiplied by the square root of minus one. And it has a remarkable effect: if a particle travels with the speed of light, then the interval between any two events along its world-line is zero.

Think of a photon, a particle of light. It travels, of course, at the speed of light. As one year of time passes, it travels one light-year. The sum of 1 and 1 is 2, but that's not how you get the interval. The interval is the difference 1 - 1, which is 0. So the interval is related to the apparent rate of passage of time for a moving observer. The faster an object moves, the slower time on it appears to pass. This effect is called time dilation. As you travel closer and closer to the speed of light, the passage of time, as you experience it, slows down. If you could travel at the speed of light, time would be frozen. No time passes on a photon.

In Newtonian physics f particles that move when no forces are acting follow straight lines. Straight lines minimise the distance between points. In relativistic physics, freely moving particles minimise the interval, and follow geodesics. Finally, gravity is incorporated, not as an extra force, but as a distortion of the structure of spacetime, which changes the size of the interval and alters the shapes of geodesics. This variable interval between nearby events is called the metric of spacetime.

The usual image is to say that spacetime becomes `curved', though this term is easily misinterpreted. In particular, it doesn't have to be curved round anything else. The curvature is interpreted physically as the force of gravity, and it causes light cones to deform.

One result is `gravitational lensing', the bending of light by massive objects, which Einstein discovered in 1911 and published in 1915. He predicted that gravity should bend light by twice the amount that Newton's Laws imply. In 1919 this prediction was confirmed, when Sir Arthur Stanley Eddington led an expedition to observe a total eclipse of the Sun in West Africa. Andrew Crommelin of Greenwich Observatory led a second expedition to Brazil. The expeditions observed stars near the edge of the Sun during the eclipse, when their light would not be swamped by the Sun's much brighter light. They found slight displacements of the stars' apparent positions, consistent with Einstein's predictions. Overjoyed, Einstein sent his mum a postcard: `Dear Mother, joyous news today ... the English expeditions have actually demonstrated the deflection of light from the Sun.' the Times ran the headline: REVOLUTION IN SCIENCE. NEW THEORY OF THE UNIVERSE. NEWTONIAN IDEAS OVERTHROWN. Halfway down the second column was a subheading: SPACE `WARPED'. Einstein became an overnight celebrity.

It would be churlish to mention that to modern eyes the observational data are decidedly dodgy - there might be some bending, and then again, there might not. So we won't. Anyway, later, better experiments confirmed Einstein's prediction. Some distant quasars produce multiple images when an intervening galaxy acts like a lens and bends their light, to create a cosmic mirage.

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 geodesics 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.

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-forall 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 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.