Science of Discworld

THE SHAPE OF THINGS

WHEN WIZARDS FIND A NEW THING, THEY PLAY WITH IT. So do scientists. They play with ideas so wild that often they seem to defy common sense, and then they insist that those ideas are right, and common sense isn't. They often make out a sur­prisingly good case. Einstein once said something nasty about common sense being akin to nonsense, but he went too far. Science and common sense are related, but indirectly. Science is something like a third cousin of common sense twice removed. Common sense tells us what the universe seems like to creatures of our particular size, habits, and disposition. For instance, common sense tells us that the Earth is flat. It looks flat, leaving out the hills, valleys, and other bumps and dents ... If it wasn't flat, things ought to roll around or fall off. Despite this, the Earth isn't flat. On Discworld, in contrast, the relation between common sense and reality is usu­ally very direct indeed. Common sense tells the wizards of Unseen University that Discworld is flat, and it is. To prove it, they can go to the Edge, as Rincewind and Twoflower do in The Colour of Magic, and watch stuff disappearing over it in Rimfall: 'The roar­ing was louder now. A squid bigger than anything Ricewind had seen before broke the surface a few hundred yards away and thrashed madly with its tentacles before sinking away ... They were running out of world.' Then they can be trapped in the Grcumfence, a ten thousand mile long net set just below the Edge, one tiny bit of which is patrolled by Tethis the sea troll. And they can peer over the edge: '... the scene beneath him flipped into a whole, new, terrifying perspective. Because down there was the head of an elephant as big as a reasonably-sized continent... Below the elephant there was nothing but the distant, painful disc of the sun. And, sweeping slowly past it, was something that for all its city-sized scales, its crater-pocks, its lunar cragginess, was indu­bitably a flipper.'

It is widely imagined that ancient people thought the Earth was flat, for all those obvious commonsense reasons. Actually, most ancient civilizations that left records seem to have worked out that the Earth has to be round. Ships came back from invisible lands over the horizon and, in the sky, a round sun and a round moon were a definite clue ...

That's where science and common sense overlap. Science is common sense applied to evidence. Using common sense in that manner, you often come to conclusions that are very different from the obvious common sense assumptions that because the universe appears to behave in some manner, then it really does. Of course it also helps to realize that if you live on a very big sphere, it's going to look pretty flat for quite a long way off. And if gravity always points towards the middle of the sphere, then things don't actually roll around or fall off. But those are refinements.

Around 250 BC a Greek called Eratosthenes tested the theory that the Earth is a sphere, and he even worked out just how big that sphere is. He knew that in the city of Syene, present-day Aswan in Egypt, the midday sun could be seen reflected in the bottom of a well. (This would not work in Ankh-Morpork, where the well-water is often more solid than the well that surrounds it.) Eratosthenes threw in a few other simple facts and got back a lot more than he'd bargained for.

It's a matter of geometry. The well was dug straight down. So the Sun at Syene had to be straight up, dead overhead. But in Eratosthenes' home city of Alexandria, in the Nile delta, that didn't happen. At midday, when the sun was at its highest, Eratosthenes cast a definite shadow. In fact, he estimated that at noon the angle between the Sun and the vertical was just over 7°, near enough 1/50 of 360°. Then came the leap of deduction. The Sun is in the same place wherever you observe it from. On other grounds, it was known that the Sun had to be a long way away from the Earth, and that meant that the Sun's rays that hit the ground in Alexandria were very nearly parallel to those that went down the well in Syene. Eratosthenes reasoned that a round Earth would explain the differ­ence. He deduced that the distance from Syene to Alexandria must be 1/50 of the circumference of the Earth. But how far was that?

On such occasions it pays to be familiar with the camel-herders. Not just because the greatest mathematician in the world is the camel called You Bastard, as it is on Discworld (see Pyramids), but because the camel trains from Alexandria to Syene took 50 days to make the trip, at an average speed of 100 stadia per day. So the dis­tance from Alexandria to Syene was 5,000 stadia, and the circumference of the Earth was 250,000 stadia. The stadium was a Greek measure of distance, and nobody knows how long it was. Scholars think it was 515 feet (157 m), and if they're right, Eratosthenes' value was 24,662 miles (39,690 km). The true value is about 24,881 miles (40,042 km), so Eratosthenes got amazingly close. Unless, sorry, but we're incorrigibly suspicious, the schol­ars worked backwards from the answer.

It is here that we encounter another feature of scientific reason­ing. In order to make comparisons between theory and experiment, you have to interpret the experiment in terms of your theory. To clarify this point, we recount the story of Ratonasticthenes, an early relative of Cut-me-own-throat Dibbler, who proved that the Discworld was round (and even estimated its circumference). Ratonasticthenes noticed that at midday in the Ramtops the Sun was overhead, whereas in Lancre, some 1000 miles away, it was at 84° to the vertical. Since 84° is roughly a quarter of 360°, Ratonasticthenes reasoned that the Discworld is round, and the dis­tance from the Ramtops to Ankh-Morpork is one-quarter of the circumference. That puts the circumference of this spherical Discworld at 4,000 miles (6,400 km). Unfortunately for this theory, it was known on other grounds that Discworld is some 10,000 miles (16,000 km) from rim to rim. Still, you can't let an awkward fact get in the way of a good theory, and Ratonasticthenes went to his grave believing that it was a small world after all.

His error was to interpret perfectly good observational data in terms of a flawed theory. Scientists repeatedly return to established theories to test them in new ways, and tend towards testiness with those priests, religious or secular, who know the answers already -whatever the questions are. Science is not about building a body of known 'facts'. It is a method for asking awkward questions and sub­jecting them to a reality-check, thus avoiding the human tendency to believe whatever makes us feel good.

From the earliest times, humans have been interested not just in the shape of the world, but in the shape of the universe. To begin with, they probably thought that these were the same question. Then they worked out, using roughly the same sort of geometry as Eratosthenes, that those lights in the sky were a very long way away. They came up with an amazing range of myths about the sun-god's fiery chariot and so on, but after the Babylonians got the idea of making accurate measurements, their theories started to lead to sur­prisingly good predictions of things like eclipses and the motion of the planets. By the time of Ptolemy (Claudius Ptolemaeus, AD 100-160) the best model of planetary motion involved a series of 'epicycles', the planets moved as if they were rotating round cir­cles whose centres rotated round other circles whose centres rotated round ...

Isaac Newton replaced this theory, and its more accurate succes­sors, with a rule, the law of gravity; it describes how each body in the universe attracts every other body. It explained Johannes Kepler's discovery that planetary orbits are ellipses, and in the full­ness of time it explained a lot of other things too.

After a few centuries of stunning success, Newton's theory ran into its first big failure: it made incorrect predictions about the orbit of Mercury. The place in its orbit at which Mercury came closest to the sun didn't move quite the way Newton's law pre­dicted. Einstein came to the rescue with a theory based not on attractive forces, but on geometry, on the shape of spacetime. This was the celebrated Theory of Relativity. The theory came in two flavours: Special Relativity and General Relativity. Special Relativity is about the structure of space, time, and electromagnet-ism; General Relativity describes what happens when you throw in gravity too.

The main point to appreciate is that 'Relativity' is a silly name. The whole point of Special Relativity is not that 'everything is rel­ative', but that one particular thing, the speed of light, is unexpectedly absolute. The thought experiment is well known. If you're travelling in a car at 50 mph (80 kph) and you fire a gun for­wards, so that the bullet moves at 500 mph (800 kph) relative to the car, then it will hit a stationary target at a speed of 550 mph (880 kph), adding the two components. However, if instead of firing the gun you switch on a torch, which 'fires' light at a speed of 670,000,000 mph (186,000 mps or 300,000 kps), then that light will not hit the stationary target at a speed of 670,000,050 mph. It will hit it at 670,000,000 mph, exactly the same speed as if the car had been stationary.

There are practical problems in staging that experiment, but less graphic and dangerous ones have indicated what the result would be.

Einstein published Special Relativity in 1905, along with the first serious evidence for quantum mechanics and a ground-break­ing paper on diffusion. A lot of other people, among them the Dutch physicist Hendrik Lorentz and the French mathematician Henri Poincare, were working on the same idea, because electro-magnetism didn't entirely agree with Newtonian mechanics. The conclusion was that the universe is a lot weirder than common sense tells us, although they probably didn't use that actual word. Objects shrink as they approach the speed of light, time slows down to a crawl, mass becomes infinite ... and nothing can go faster than light. Another key idea was that space and time are to some extent interchangeable. The traditional three dimensions of space plus a separate one for time are merged into a single unified spacetime with four dimensions. A point in space becomes an event in spacetime.

In ordinary space, there is a concept of distance. In Special Relativity, there is an analogous quantity, called the interval between events, which is related to the apparent rate of flow of time. The faster an object moves, the slower time flows for an observer sitting on that object. This effect is called time dilation.

If you could travel at the speed of light, time would be frozen.

One startling feature of relativity is the twin paradox, pointed out by Paul Langevin in 1911. Again, it is a classic illustration. Suppose that Rosencrantz and Guildenstern are born on Earth on the same day. Rosencrantz stays there all his life, while Guildenstern travels away at nearly lightspeed, and then turns round and comes home again. Because of time dilation, only one year (say) has passed for Guildenstern, whereas 40 years have gone by for Rosencrantz. So Guildenstern is now 39 years younger than his twin brother. Experiments carrying atomic clocks around the Earth on jumbo jets have verified this scenario, but aircraft are so slow compared to light that the time difference observed (and pre­dicted) is only the tiniest fraction of a second.

So far so good, but there's no place yet for gravity. Einstein racked his brains for years until he found a way to put gravity in: let spacetime be curved. The resulting theory is called General Relativity, and it is a synthesis of Newtonian gravitation and Special Relativity. In Newton's view, gravity is a force that moves particles away from the perfect straight line paths that they would otherwise follow In General Relativity, gravity is not a force: it is a distortion of the structure of spacetime. The usual image is to say that space-time 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 rays to bend. One result is 'gravitational lensing', the bending of light by massive objects, which Einstein discovered in 1911 and published in 1915. The effect was first observed during an eclipse of the Sun. More recently it has been discovered that some distant quasars produce multiple images in telescopes because their light is lensed by an intervening galaxy.

Einstein's theory of gravity ousted Newton's because it fitted observations better, but Newton's remains accurate enough for many purposes, and is simpler, so it is by no means obsolete. Now it's beginning to look as if Einstein may in turn be ousted, possibly by a theory that he rejected as his greatest mistake.

In 1998 two different observations called Einstein's theory into question. One involved the structure of the universe on truly mas­sive scales, the other happened in our own backyard. The first has survived everything so far thrown at it; the second can possibly be traced to something more prosaic. So let's start with the second curious discovery.

In 1972 and 1973 two space probes, Pioneer 10 and 11, were launched to study Jupiter and Saturn. By the end of the 1980s they were in deep space, heading out of the known solar system. There has long been a belief, a scientific legend waiting to happen, that beyond Pluto there may be an as yet undiscovered planet, Planet X. Such a planet would disturb the motions of the two Pioneers, so it was worth tracking the probes in the hope of finding unexpected deviations. John Andersen's team found deviations, all right, but they didn't fit Planet X, and they didn't fit General Relativity either. The Pioneers are coasting, with no active form of propul­sion, so the gravity of the Sun (and the much weaker gravity of the other bodies of the known solar system) pulls on them and gradu­ally slows them down. But the probes were slowing down a tiny bit more than they should have been. In 1994 Michael Martin sug­gested that this effect had become sufficiently well established that it cast doubt on Einstein's theory, and in 1998 Anderson's team reported that what was observed could not be explained by such effects as instrument error, gas clouds, the push of sunlight, or the gravitational pull of outlying comets.

Three other scientists quickly responded by suggesting other things that might explain the anomalies. Two wondered about waste heat. The Pioneers are powered by onboard nuclear generators, and they radiate a small amount of surplus heat into space. The pressure of that radiation might slow the craft down by the observed amount. The other possible explanation is that the Pioneers may be venting tiny quantities of fuel into space. Anderson thought about these explanations and found problems with them both.

The strangest feature of the observed slowing down is that it is precisely what would be predicted by an unorthodox theory sug­gested in 1983 by Mordehai Milgrom. This theory changes not the law of gravity, but Newton's law of motion: force equals mass times acceleration. Milgrom's modification applies when the acceleration is very small, and it was introduced in order to explain another gravitational puzzle, the fact that galaxies do not rotate at the speeds predicted by either Newton or Einstein. This discrepancy is usually put down to the existence of 'cold dark matter' which exerts a grav­itational pull but can't be seen in telescopes. If galaxies have a halo of cold dark matter then they will rotate at a speed that is inconsis­tent with the matter in the visible portions. A lot of theorists dislike cold dark matter (because you can't observe it directly, that's what 'cold dark' means) and Milgrom's theory has slowly gained in pop­ularity. Further studies of the Pioneers may help decide.

The other discovery is about the expansion of the universe. The universe is getting bigger, but it now seems that the very distant universe is expanding faster than it ought to. This startling result -confirmed by later, more detailed studies, comes from the Supernova Cosmology project headed by Saul Perlmutter and its arch-rival High-Z Supernova Search Team headed by Brian Schmidt. It shows up as a slight bend in a graph of how a distant supernova's apparent brightness varies with its red shift. According to General Relativity, that graph ought to be straight, but it's not. It behaves as if there is some repulsive component to gravity which only shows up at extremely long distances, say half the radius of the universe. A form of antigravity, in fact.

Curiously, Einstein originally included a repulsive force of this kind in his relativistic equations for gravity: he called it the cosmo-logical constant. Later he changed his mind and threw the cosmological constant out, complaining that he'd been foolish to include it in the first place. He died thinking it was a blemish on his record, but maybe his original intuition was spot on after all.

There is also a possible link to the other deep physical theory, quantum mechanics. At first this looked unlikely. If there is an antigravity effect, then it should stem from Vacuum energy', a form of energy that, if it exists, is stored in empty space ... (As we write this, we can picture Ridcully's expression. We shall have to ignore it. This isn't something sensible, like magic. This is science. Empty space can be full of interest.)

However, quantum theory predicts that if vacuum energy exists in today's universe, then it would produce an antigravity effect 10¹¹⁹ (1 followed by 119 zeros) times as big as what's observed. Although astronomers are accustomed to larger experimental errors than you find in most other sciences, this is too much for even them to swal­low. But late in 1998 Robert Matthews wondered whether the antigravity effect might come from a relic of the vacuum energy of an earlier phase of the universe. His idea is related to a sixty-year-old piece of speculation by Paul Dirac, one of the founders of quantum theory. Dirac noticed a strange coincidence. The electro­magnetic force between a proton and an electron is 10⁴⁰ (1 followed by 40 zeros) times as great as the gravitational force between them. The age of the universe is also 10⁴⁰ times as great as the time it takes light to cross one atom. It's not hard to come up with numerologi-cal accidents of this kind, but Dirac had a hunch that this one might indicate some deep connection between the expansion of the uni­verse and the microscopic quantum realm. Now Matthews has come up with a possible explanation of the coincidence, and it fits the antigravity effect.

According to the Big Bang theory, the early history of the uni­verse involves a number of 'phase transitions', dramatic changes of state which result in big qualitative changes in how the universe works. The earliest one occurred when the strong nuclear force sep­arated from the electromagnetic forces and the weak nuclear force. The last in this series of phase transitions was the quark-hadron transition, in which quarks grouped together to produce the more familiar protons and neutrons. If the universe has somehow retained the vacuum energy from this phase transition, then it will exhibit an antigravity effect of just the right size. So these curious observations may be telling us something rather curious about the early universe.