The science of discworld.., p.24
The Science of Discworld IV, page 24
To be specific, it is Poincaré’s dodecahedral space.
To make a dodecahedral space, you start with a dodecahedron. This is a solid with twelve faces, each a regular pentagon; like a football without the hexagons. Then you glue opposite faces together – something that is not possible with a real dodecahedron. Mathematically, there is a way to pretend that distinct faces are actually the same, without bending the thing to join them together, as we saw for the flat torus; topologists, however, insist on calling it ‘gluing’.
The dodecahedral space is an elaborate variation on a flat torus. Recall that we make a flat torus by taking a square and gluing opposite edges together. To get the dodecahedral space, which is not a surface but a three-dimensional object, we take a dodecahedron and glue opposite faces together. The result is a three-dimensional topological space. It has no boundary, just like a torus, and for the same reason: anything that is in danger of falling out through a face reappears inside at the opposite one, so there’s no way out. It has finite size. And, like a hypersphere, it has no holes, so if you are a slightly naive topologist you might be tempted to think it passes all the tests needed to be a hypersphere – but it isn’t a hypersphere, not even topologically.
Poincaré devised his dodecahedral space as a piece of pure mathematics, exposing a limitation of the topological methods available in his day – one that he set out to remedy. But in 2003 the dodecahedral space acquired brief notoriety and a potential application to cosmology when NASA’s Wilkinson Microwave Anisotropy Probe (WMAP) satellite was measuring fluctuations in the cosmic microwave background, a persistent hiss detected by radio telescopes that is interpreted as a relic of the Big Bang. The statistics of these tiny irregularities provide information about how matter clumped together in the early universe, acting as a seed from which stars and galaxies formed. WMAP can see far enough in space to, in effect, see back in time to about 380,000 years after the Big Bang.
At the time, most cosmologists thought that the universe was infinite. (Although this conflicts with the standard description of the Big Bang, there are ways to accommodate it, and ‘universes all the way out’ has the innate appeal already noted for ‘universes all the way back’ – which, ironically, is not what the Big Bang indicates.) However, the WMAP data suggested that the universe is finite. An infinite universe ought to support fluctuations of all sizes, but the data did not show any large waves. As a report in Nature said at the time, ‘you don’t see breakers in your bathtub’. The detailed data provided further clues about the likely shape of our breakerless bathtub universe. Working out the statistics of the fluctuations for a variety of potential shapes, the mathematician Jeffrey Weeks noticed that the dodecahedral space fitted the data very well, without any special pleading. Jean-Pierre Luminet’s group published an analysis showing that if this were correct, the universe would have to be about 30 billion light years across.fn6 This theory has since fallen out of favour thanks to further observations, but it was fun while it lasted.
We human ants can use another trick to infer the shape of space. If the universe is finite, some rays of light will eventually return to their point of origin. If you could look along one of these ‘closed geodesics’ (a geodesic is a shortest path) with a sufficiently high-powered telescope, and if light travelled infinitely fast, you would see the back of your own head. Taking the finite speed of light into account, patterns should occur in the cosmic microwave background, forming matching circles in the sky. The way these circles are arranged would provide information about the topology of space. Cosmologists and mathematicians have tried to find such circles, so far with no convincing successes. If the universe is finite but too big, we wouldn’t be able to see far enough to spot the circles anyway.
So the current answer to the question ‘what shape is the universe?’ is very simple. We don’t know. We don’t know whether it’s a hypersphere or something more elaborate. The universe is too big for us to observe it all, and our current understanding of cosmology, indeed of fundamental physics, wouldn’t be up to the task even if we could.
Some of the difficulties surrounding cosmology stem from a mix-and-match approach in which relativity is invoked at some stages and quantum mechanics at others, without recognising that they contradict each other. Theorists are reluctant to discard the tools they are accustomed to, even when those tools don’t seem to be working. But the shape of the universe is a problem that really requires a combination of these two great physical theories. And that brings us to the search for a unified field theory, or Theory of Everything, to which Einstein devoted many years of thought – without success. Somehow, relativity and quantum mechanics have to be modified to produce a consistent theory that agrees with each of them in the relevant domain.
Today’s front runner is string theory, which replaces point particles by tiny multidimensional shapes, as we discussed in The Science of Discworld III. Some versions of string theory require space to be nine-dimensional, so spacetime has to be ten-dimensional. The extra six dimensions of space are thought either to be curled up so tightly that we don’t notice them, or inaccessible to humans – in the same way that A. Square could not travel out of Flatland unaided, but needed a shove from the Sphere to push him into the third dimension. The formulations of string theory currently in vogue also introduce new ‘super-symmetry’ principles, predicting the existence of a host of new ‘sparticles’ to match the known particles. So an electron is paired with a selectron, and so on. So far, however, this prediction has not been confirmed. The LHC has looked for sparticles, but so far it has found precisely none of them.
One of the latest unification attempts, refreshingly different from most that have gone before, takes us neatly back to Flatland. The idea, common in mathematics and often fruitful, is to take inspiration from a cut-down toy problem. If unifying relativity and quantum mechanics is too hard using three-dimensional space, why not simplify the problems by looking at the non-physical but mathematically informative case of two-dimensional space? Plus one of time, naturally. It’s clear enough where to start. In order to unify two theories, you have to have two theories to unify. So what would gravity look like in Flatland, and what would quantum mechanics look like in Flatland? We hasten to add that Flatland, here, need not be A. Square’s Euclidean plane. Any two-dimensional space, any surface, would do. Indeed, other topologies are essential to get anything interesting.
It’s straightforward to write down sensible analogues of Einstein’s field equations when space is a surface. It’s very close to what Gauss did when he started the whole thing off, and his ant would have no trouble in devising the right equations, since they’re all about curvature. There are obvious analogies to follow; all you do is replace the number three by the number two at key points. In Roundworld, the Polish physicist Andrzej Staruszkiewicz wrote such equations down in 1963.
It turns out that gravity in two dimensions differs significantly from gravity in three. In three dimensions relativity predicts the existence of gravitational waves, which propagate at the speed of light. But there are no gravitational waves in two dimensions. In three dimensions, relativity predicts that any mass bends space into a rounded bump, so that anything that passes nearby will follow a curved path, as if it were attracted by Newtonian gravity. And an object that was at rest will fall into the gravitational well of the mass concerned. In two dimensions, however, gravity bends space into a cone. Moving bodies are deflected, but bodies at rest simply remain at rest. In three dimensions massive bodies collapse under their own gravitation to form black holes. In two dimensions, this is impossible.
These differences are things we can live with, but in three dimensions, gravitational waves are a useful way to link relativity to quantum theory. The absence of gravitational waves in two dimensions causes headaches, because it means there is nothing to quantise – nothing to use as a starting point for a quantum-mechanical formulation. Gravity should correspond to hypothetical particles called gravitons, and in quantum theory particles have ghostly companions – waves. No waves, no gravitons. But in 1989 Edward Witten, one of the architects of string theory, ran into other quantum problems involving fields that do not propagate waves. Two-dimensional gravity is like that, and it opened his eyes to a missing ingredient.
Topology.
Even when gravity can’t travel as a wave, it can have a huge effect on the shape of space. Witten’s experience with topological quantum field theories, where just this ingredient arises, suggested a way forward. The humble torus, in many ways the simplest non-trivial topological space, plays a key role. We’ve mentioned the flat torus, in which we glue together the opposite edges of a square. Squares are nice because they can be fitted with grids of smaller squares, which have a quantum feel to them because they are discrete – they come in tiny lumps. But you can make flat tori from other shapes too, namely parallelograms.
The shape of the parallelogram can be captured by a number called the modulus, which distinguishes long thin parallelograms from short fat ones. A different modulus gives a different torus. Although the tori obtained in this manner are flat, they have different metrics. They can’t be mapped into each other while keeping all distances the same. The effect of gravity in Torusland is not to create gravitons: it is to change the modulus, the shape of space.
Steven Carlipp has shown that in Torusland, there is an analogue of the Big Bang. But it doesn’t start with a point singularity. Instead, it begins as a circle: a torus with modulus zero. As time passes, the modulus increases, and the circle expands into a torus. Initially this looks like a bicycle tyre, and corresponds to a long thin parallelogram; it is heading towards a square, the standard model for a flat torus, which when curled up looks more like a bagel. So the long-term goal of the Flatland Big Bang turns out to be A. Square. Crucially, Carlipp quantised this entire process; that is, he formulated a quantum-mechanical analogue. That let theoretical physicists explore the relationship between quantum theory and gravity in a precise mathematical context.
Torusland sheds a great deal of light on the process of quantising a gravitational theory. One apparent casualty of this process, however, is time. The quantum wave function of Torusland does not involve time at all.
In The Science of Discworld III Chapter 6, we discussed Julian Barbour’s The End of Time, which proposes that time does not exist in a quantum world because there is only one universal wavefunction, not involving time. The book was widely interpreted as telling us that time is an illusion. ‘There can only be once-and-for-all probabilities,’ Barbour wrote. We argued that alongside the universal wavefunction, our universe has another basic quantum-theoretic feature, which describes how likely transitions between different states are. These transition probabilities show that some states are closer together than others, and that lets us arrange the events in a natural order, restoring a sensible notion of time.
Torusland supports this idea, because it has several sensible notions of time, even though its quantum wavefunction is timeless. Time can be measured using Torusland’s equivalent of GPS satellites, by using the lengths of curves between its version of the Big Bang and ‘now’, or by the current size of the universe. Torusland is not timeless at all. You just have to look at it in the right way. In fact, Torusland time leads to an intriguing thought: perhaps time is a consequence of gravity.
Another idea that Torusland casts doubt on is the holographic principle. This says that the quantum state of the entire observable universe can be ‘projected’ onto any black hole’s event horizon – the point of no return from which nothing can escape – so the universe’s three spatial dimensions can be reduced to just two. It’s like taking a photograph, with the startling property that the photograph faithfully represents all aspects of reality. In Roundworld, if someone shows you a photo of a field with a dozen sheep lying down, you can’t tell whether there are lambs hiding behind some of the sheep. But in this event-horizon photo of the universe, nothing can be hidden. The behaviour in two dimensions corresponds perfectly to that in three. The laws of physics change, but everything matches up.
This is a bit like the way a two-dimensional hologram creates a three-dimensional image, which is why this idea is called the holographic principle. It suggests that not only is the dimension of the universe an open question: it may not be well defined – the answers ‘two’ and ‘three’ may both be true at the same time. This idea has led to some advances in the way string theory represents gravity, and also to articles in the press stating ‘You are a hologram!’
Physicists began to suspect that a similar principle works in any number of dimensions. But it turns out that in Torusland, there is no holographic principle. A. Square may be flat, but he’s not a hologram. So maybe we’re not holograms either. Which would be nice.
Some even more radical ideas about the shape of our universe have just surfaced, threatening to overturn many deep-seated assumptions in cosmology. Instead of being a gigantic hypersphere, or a flat Euclidean space, the universe might be more like an etching by the Dutch artist Maurits Escher.
Welcome to the Escherverse.
A hypersphere is the iconic surface with constant positive curvature. There is also an iconic surface of constant negative curvature, called the hyperbolic plane. It can be visualised as a circular disc in the usual Euclidean plane, equipped with an unusual metric, in which the unit of measurement shrinks the closer you get to the boundary. Escher based some of his etchings on the hyperbolic plane. A famous one, which he called ‘Circle Limit IV’ but is usually referred to as ‘angels and devils’, tiles the disc with black devils and white angels. Near the middle these appear quite large; as they approach the boundary they shrink, so that in principle there would be infinitely many of them. In the metric of the hyperbolic plane, all devils are the same size, and so are all angels.
String theory tries to unify the three quantum-mechanical forces (weak, strong and electromagnetic) with the relativistic force of gravity, and gravity is all about curvature. So curvature plays a key role in string theory. However, attempts to marry string theory to relativistic cosmology tend to come to grief, because string theory works best in spaces with negative curvature, whereas positive curvature works better for the cosmos. Which is a nuisance.
At least, that’s what everyone thought.
But in 2012 Stephen Hawking, Thomas Hertog and James Hartle discovered that they could use a version of string theory to write down a quantum wavefunction for the universe – indeed, for all plausible variations on the universe – using a space with constant negative curvature. This is the Escherverse. It’s terrific mathematics, and it disproves some widely believed assumptions about the curvature of spacetime. Whether it will also work out as physics remains to be seen.
So what have we learned? That the shape of our universe is intimately related to the laws of nature, and its study sheds some light – and a lot more darkness – on possible ways to unify relativity and quantum theory. Mathematical models like Torusland and the Escherverse have opened up new possibilities by showing that some common assumptions are wrong. But despite all of these fascinating developments, we don’t know what shape our universe is. We don’t know whether it is finite or infinite. We don’t even know for sure what dimension it is, or even whether its dimension can be pinned down uniquely.
Like A. Square, trapped in Flatland, we are unable to step outside our world and view it unobstructed. But, also like him, we can learn a lot about the world despite that. On Discworld, creatures from the dungeon dimensions are only an incantation away; in Flatland a helpful Sphere may pop into view to help the story along. But Roundworld doesn’t run on narrativium, and an extra-universal visitor from hidden dimensions seems unlikely.
So we are stuck with our own resources: imagination, ingenuity, logic and respect for evidence. With these, we can hope to infer more about our universe. Is it finite or infinite? Is it four-dimensional or eleven-dimensional? Is it round, flat or hyperbolic?
For all we know right now, it might be banana-shaped.
fn1 Yes, two Abbotts. His father was Edwin Abbott. So was his son.
fn2 Abbott never said what the ‘A’ stood for. One theory is that A2 = AA = Abbott Abbott. In Ian’s modern sequel Flatterland it is ‘Albert’. Google ‘Albert Square’.
fn3 Some mathematicians think he did, but the physicists didn’t notice because he wasn’t a physicist.
fn4 So do the wizards: see Unseen Academicals.
fn5 By the 2006 World Cup it was made from 14 panels: six dumbbell-shaped ones and eight like the Isle of Man’s triple running-legs emblem. The underlying symmetry was again that of a cube. If you think that analysing symmetries of footballs is nerdy, look up the literature on symmetries of golf balls.
fn6 J-P Luminet, Jeffrey R. Weeks, Alain Riazuelo, Roland Lehoucq and Jean-Philippe Uzan, Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background, Nature 425 (2003) 593.
SEVENTEEN
* * *
THE WIZARD FORMERLY KNOWN AS THE DEAN
The black gallery wasn’t as black or foreboding as Marjorie had expected; it was just filled with pictures of long-deceased people with no indication of how they had become deceased, these facts lost now to memory as well as to life.
The wizards went into a huddle, and she heard the Archchancellor say, ‘Look! We have always known we were not your average planet; after all, we have sometimes passed other more ordinary planets as the turtle has moved, and often as you know by occult ways and means. I think the opposition will try to say that we are somehow on a freak world. I am debating with myself whether or not to allow them to feel that is the way forward. What do you say, Mister Stibbons?’
