The Science of Discworld

by Terry Pratchett

  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.

  Recent work seems to have confirmed this remarkable discovery. But — as always — ingenious scientists have come up with alternative explanations. In 2001 Csaba Csáki, John Terning, and Nemanja Kaloper put forward a totally different theory to explain the observations. They suggest that the light from distant supernovas is dimmer than expected because some of the particles of light — photons — are changing into something else. Specifically, they are changing into 'axions', hypothetical particles predicted by several of the currently fashionable quantum-mechanical theories of particle physics. Axions are not expected to interact much with other matter, which makes it hard to detect them; but if they have a very small but non-zero mass, about one sextillionth of that of an electron, then they will interact with intergalactic magnetic fields. This interaction would convert a small fraction of photons into axions, and that would account for the missing light. In fact, the most distant supernovas could lose one third of their photons this way.

  It is a sobering thought that such a tiny a modification of known physics, by introducing a particle whose mass ought to be negligible, could have such a big effect. At any rate, either gravity is not as we thought, or axions exist (as expected) and have mass (not as expected). Or there's a third reason for the observations, which no one has yet thought of.

  One theory of the repulsive force is an exotic form of matter, 'quintessence'.* This is a form of vacuum energy that pervades all of space, and exerts negative pressure. (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.) Curiously, Einstein originally included a repulsive force of this kind in his relativistic equations for gravity: he called it the cosmological 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.

  Unless axions exist and have mass, of course.

  In Einstein's approach to the cosmological constant, quintessence is effectively spread uniformly throughout space. But suppose it isn't? Ordinary matter is clumpy, not uniform. David Santiago has pointed out that if quintessence is clumpy too, then Einstein's Equations predict that the universe could contain 'anti-Black Holes' that repel matter instead of swallowing it. These are not quite the same as hypothetical White Holes, time-reversed Black Holes, which spit matter out. However, it's not yet clear that anti-Black Holes can be stable. Ordinary matter is clumpy because gravity is an attractive force — it likes to create clumps. Antigravity is a repulsive force, and by analogy it ought to destroy clumps. If that argument is right, then anti-Black Holes are unstable, and would not be able to form in the first place. They would be mathematical solutions of Einstein's Equations, but not ones that could be physically realised. Until somebody does the necessary calculations, we can't be sure.

  'Most civilizations' is admittedly not the same as 'most people'. 'Most people' throughout the history of the planet have not needed to concern themselves with what shape the world is, provided it supports, somewhere, the next me
al.

  This word, meaning 'fifth essence', originally referred to a fifth 'element' after earth, air, fire and water. On Discworld this role is played by surprise.

  ELEVEN

  NEVER TRUST A CURVED UNIVERSE

  PONDER STIBBONS HAD SET UP A DESK a little separate from the others and surrounded it with a lot of equipment, primarily in order to hear himself think.

  Everyone knew that stars were points of light. If they weren't, some would be visibly bigger than others. Some were fainter than others, of course, but that was probably due to clouds. In any case their purpose, according to established Discworld law, was to lend a little style to the night.

  And everyone knew that the natural way for things to move was in a straight line. If you dropped something, it hit the ground. It didn't curve. The water fell over the edge of the world, drifting sideways just a tiny bit to make up for the spin, but that was common sense. But inside the Project, spin was everything. Everything was bent. Archchancellor Ridcully seemed to think this was some sort of large-scale character flaw, akin to shuffling your feet or not owning up to things. You couldn't trust a universe of curves. It wasn't playing a straight bat.

  At the moment Ponder was rolling damp paper into little balls. He'd had the gardener push in a large stone ball that had spent the last few hundred years on the university's rockery, relic of some ancient siege catapult. It was about three feet across.

  He'd hung some paper balls of string near it. Now, glumly, he threw others over it and around it. One or two did stick, admittedly, but only because they were damp. He was in the grip of some thought, You had to start with what you were certain of. Things fell down. Little things fell down on to big things. That was common sense.

  But what would happen if you had two big things all alone in the universe?

  He set up two balls of ice and rock, in an unused corner of the Project, and watched them bang into each other. Then he tried with ball of different sizes. Small ones drifted towards big ones but, oddly enough, the big ones also drifted slightly towards the small ones.

  So ... if you thought that one through ... that meant that if you dropped a tennis ball to the ground it would certainly go down, but in some tiny, immeasurable way the world would, very slightly, come up.

  And that was insane.

  He also spent some time watching clouds of gas swirl and heat in the more distant regions of the Project. It was all so ... well, godless.

  Ponder Stibbons was an atheist. Most wizards were. This was because UU had some quite powerful standing spells against occult interference, and knowing that you're immune from lightning bolts does wonders for an independent mind. Because the gods, of course, existed. Ponder wouldn't even attempt to deny it. He just didn't believe in them. The god currently gaining popularity was Om, who never answered prayers or manifested himself. It was easy to respect an invisible god. It was the ones that turned up everywhere, often drunk, that put people off.

  That's why, hundreds of years before, philosophers had decided that there was another set of beings, the creators, that existed independently of human belief and who had actually built the universe. They certainly couldn't have been gods of the sort you got now, who by all accounts were largely incapable of making a cup of coffee.

  The universe inside the Project was hurtling through its high-speed time and there was still nothing in there that was even vaguely homely for humans. It was all too hot or too cold or too empty or too crushed. And, distressingly, there was no sign of narrativium.

  Admittedly, it has never been isolated on Discworld either, but its existence had long ago been inferred, as the philosopher Lye Tin Wheedle had put it: 'in the same way that milk infers cows'. It might not even have a discrete existence. It might be a particular way in which every other element spun through history, something that they had but did not actually possess, like the gleam on the skin of a polished apple. It was the glue of the universe, the frame that held all the others, the thing that told the world what it was going to be, that gave it purpose and direction. You could detect narrativium, in fact, by simply thinking about the world.

  Without it, apparently, everything all was just balls spinning in circles, without meaning.

  He doodled on the pad in front of him:

  There are no turtles anywhere.

  'Eat hot plasma! Oh ... sorry, sir.'

  Ponder peered over his defensive screen.

  'When worlds collide, young man, someone is doing something wrong!'

  That was the voice of the Senior Wrangler. It sounded more petulant than usual.

  Ponder went to see what was going on.

  TWELVE

  WHERE DO RULES COME FROM?

  SOMETHING IS MAKING ROUNDWORLD DO STRANGE THINGS . . .

  It seems to be obeying rules. Or maybe it's just making them up as it goes along.

  Isaac Newton taught us that our universe runs on rules, and they are mathematical. In his day they were called 'laws of nature', but 'law' is too strong a word, too final, too arrogant. But it does seem that there are more or less deep patterns in how the universe works. Human beings can formulate those patterns as mathematical rules, and use the resulting descriptions to work out some aspects of nature that would otherwise be totally mysterious, and even exploit them to make tools, vehicles, technology.

  Thomas Malthus changed a lot of people's minds when he found a mathematical rule for social behaviour. He said that food grows arithmetically (1-2-3-4-5), but populations grow geometrically (1-2-4-8-16). Whatever the growth rates, eventually population will outstrip food supply: there are limits to growth.* Malthus's law shows that there are rules Down Here as well as Up There, and it tells us that poverty is not the result of evil or sin. Rules can have deep implications.

  What are rules? Do they tell us how the universe 'really' works, or do our pattern-seeking brains invent or select them?

  There are two main viewpoints here. One is fundamentalist at heart, as fundamentalist as the Taliban and Southern Baptists — indeed, as fundamentalist as the exquisitor Vorbis in Small Gods who states his position thus: ' . . . that which appears to our senses is not the fundamental truth. Things that are seen and heard and done by the flesh are mere shadows of a deeper reality.'

  Scientific fundamentalism holds that there is one set of rules, the Theory of Everything, which doesn't just describe nature rather well, but is nature. For about three centuries science seems to have been converging on just such a system: the deeper our theories of nature become, the simpler they become too. The philosophy behind this view is known as reductionism, and it proceeds by taking things to bits, seeing what the bits are and how they fit together, and using the bits to explain the whole. It's a very effective research strategy, and it's served us well for a long time. We've now managed to reduce our deepest theories to just two: quantum mechanics and relativity.

  Quantum mechanics set out to describe the universe on very small scales, subatomic scales, but then became involved in the largest scales of all, the origin of the universe in the Big Bang. Relativity set out to describe the universe on very large scales, supergalactic ones, but then became involved in the smallest scales of all, the quantum effects of gravity. Despite this, the two theories disagree in fundamental ways about the nature of the universe and what rules it obeys. The Theory of Everything, it is hoped, will subtly modify both theories in such a way that they fit seamlessly together into a unified whole, while continuing to work well in their respective domains. With everything reduced to one Ultimate Rule, reductionism will have reached the end of its quest, and the universe will be completely explained.

  The extreme version of the alternative view is that there are no ultimate rules, indeed that there are no totally accurate rules either. What we call laws of nature are human approximations to regularities that crop up in certain specialized regions of the universe — chemical molecules, galaxy dynamics, whatever. There is no reason why our formulations of regularities in molecules and regularities in g
alaxies should be part of some deeper set of regularities that explains both, any more than chess and soccer should somehow be aspects of the same greater game. The universe could perfectly well be patterned on all levels, without there being an ultimate pattern from which all the others must logically follow. In this view, each set of rules is accompanied by a statement of which areas it can safely be used to describe — 'use these rules for molecules with fewer than a hundred atoms' or 'this rule works for galaxies provided you don't ask about the stars that make them up'. Many such rules are contextual rather than reductionist: they explain why things work the way they do in terms of what is outside them.

  Evolution, especially before it was interpreted through the eyes of DNA, is one of the clearest examples of this style of reasoning. Animals evolve because of the environment in which they live, including other animals. A curious feature of this viewpoint is that to a great extent the system builds its own rules, as well as obeying them. It is rather like a game of chess played with tiles that can be used to build new bits of board, upon which new kinds of chess piece can move in new ways.

  Could the entire universe sometimes build its own rules as it proceeds? We've suggested as much a couple of times: here's a sense in which it might happen. It's hard to see how rules for matter could meaningfully 'exist' when there is no matter, only radiation — as there was at an early stage of the Big Bang. Fundamentalists would maintain that the rules for matter were always implicit in the Theory of Everything, and became explicit when matter appeared. We wonder whether the same 'phase transition' that created matter might also have created its rules. Physics might not be like that, but biology surely is. Before organisms appeared, there couldn't have been any rules for evolution.