Light from distant galaxies tells us that the universe is expanding-one of the main pieces of evidence that space, time and everything came into existence a little over 12bn years ago in the big bang. In 1998 astronomers, trying to find out whether the expansion will continue forever, or grind to a halt and reverse itself in a big crunch, discovered something much more puzzling. The expansion is speeding up. To explain this baffling acceleration, the cosmologists invented dark energy, a mysterious force that pushes the universe apart.
Does dark energy exist? No one knows. At present nothing known to physics can explain it, so something unknown to physics must be the cause. It's like something out of Star Wars.
In February this year, American cosmologists Gia Dvali and Michael S Turner put forward a different theory, one in which dark energy does not exist. Instead, gravity is leaking out of our universe into an extra dimension. With less gravity to hold the universe together, it is coming apart faster than expected. It also sounds like something out of Star Wars.
Hidden dimensions? Only in the late 20th and early 21st centuries could physicists say this kind of thing with a straight face. It is a concept associated with Victorian spiritualists, who invented the fourth dimension as a convenient place to hide everything that didn't make sense in the familiar three.
Why we need the superstring hypothesis
We spent the first half of the 20th century learning that the universe is far stranger than we imagined. Albert Einstein taught us that not only do space and time together make up a four-dimensional continuum; they also get mixed up with each other if we move fast enough-this is relativity. And Werner Heisenberg, Erwin Schr?ger and Paul Dirac discovered that on the tiniest of scales, the universe is plain weird: the quantum world, in which matter is made of waves and cats can be alive and dead at the same time.
We spent the last half of the 20th century puzzling over one gigantic discrepancy: relativity and quantum theory contradict each other. Each works well within its own domain-the very large for relativity, the very small for quantum theory. But when those domains overlap, as they do when we want to understand the early history of the universe, the combination doesn't work. And so science set off on a quest for a single theory that would unify the whole of physics into a single mathematical law. And out of that quest came a strong suspicion that the familiar three dimensions of space and a fourth of time are mere scratches on the surface of something far bigger. Could the universe be made from ten-dimensional "superstrings," maybe, with six tightly curled dimensions that are so small we never notice them? Or is the universe just a four-dimensional "brane" floating in a many-dimensional metaverse, like a skin of congealed milk on a cup of coffee?
Somewhere in that half century, physics lost contact with the world in which most of us live. However, it is worth recognising that their world may be more real than ours; the human-centred viewpoint works fine for activities like politics and art, but it may not be appropriate for a universe that operates in inhuman ways and on scales that the human mind did not evolve to contemplate.
That, however, is how humanity grows: by finding new ways to think the unthinkable. And so I offer here an essential piece of mental kit: the bluffer's guide to superstrings. But our tour of string country and braneworld must begin with a crash course in relativity and quantum mechanics.
The key to both is electricity. In 1864 James Clerk Maxwell took the physical insights of Michael Faraday and turned them into four mathematical equations that described the laws relating magnetism and electricity. Maxwell's equations showed that electromagnetic waves must exist, and that they must travel at the speed of light. What travels at the speed of light? Light. So light is an electromagnetic wave. Physics has never been the same since, and neither has human civilisation, because one form of "light" is radio waves. Maxwell's equations led directly to the discovery of radio waves, and the invention of radio, radar, and television.
Around the end of the 19th century, mathematicians noticed something strange about Maxwell's equations. The laws of physics ought to be the same in all frames of reference, but Maxwell's equations are not. To a moving observer, the laws of nature described by Maxwell are different from the laws as seen by a stationary observer. It took Einstein to puzzle out what was happening; his answer was relativity. Paradoxically, the theory of relativity does not state that everything is relative-that was what the earlier Newtonian theory said. According to Einstein, one thing is absolute: the speed of light.
The consequences are bizarre. You can't travel faster than light or send messages faster than light. No Star Wars hyperdrives. When you get very close to the speed of light, distances shrink, time slows down, and mass increases without limit. But-and here's the wonderful thing-you don't notice, because your measuring instruments also shrink, slow down, or get heavier. So the laws of physics cannot tell you whether you are moving or stationary. They can tell you whether you are accelerating, but not how fast. Experiments confirm the theory in minute detail.
Light is so familiar to us that we seldom think about how weird it is. It seems to weigh nothing, it penetrates everywhere, and it enables us to see. What is light? Electromagnetic waves. Waves in what? The space-time continuum, which is a fancy way of saying "we don't know." Early in the 20th century, the medium for the waves was thought to be the luminiferous ether. After Einstein, we understood one thing about that ether: it doesn't exist. The waves are not in anything.
This makes sense, because the next development told us that things are themselves waves. I'm a wave, you're a wave, all God's chillun' are waves. The universe is made of waves. Which are also particles. When they feel like it. Let me explain.
Light can be converted to electricity: this is the "photoelectric effect." Shine light on a suitable metal and it will emit electricity. Einstein noticed that the electricity comes in discrete packages. Indeed, electricity is the motion of tiny particles called electrons. He deduced that the same goes for light. So light waves are also particles-photons. This leads to the quantum picture of the world, in which particles are really waves, piled up in localised humps. Sometimes they behave like particles, sometimes like waves, occasionally like both at once (this used to be anathema but now turns out to be true, albeit very rarely). The motion of these "wavicles" obeys another piece of mathematics: Schr?ger's equation.
Erwin Schr?ger is famous for one other idea: his cat. According to quantum mechanics, the wavicles can interfere with each other, piling up on top of each other and reinforcing, or cancelling each other out as peak meets trough. Quantum wavicles can "superpose." Indeed, that is the natural state of affairs; only when we observe something do we force it out of some quantum superposition and into a single "pure" state. The textbooks did this with the spin of the electron, but Schr?ger wondered how it would work with a cat. In his thought experiment, a cat locked in a box can be in a superposition of the states alive and dead. When you open the box, you observe the cat and force it into either one state or the other. As Terry Pratchett noted in his comic novel Maskerade, cats aren't like that. Greebo, a cat belonging to a witch, emerges from a box in a third state: absolutely bloody furious.
Schr?ger also knew that cats aren't like that, although for different reasons. An electron is a microscopic entity, on the quantum level, and it behaves like something on the quantum level. A cat is macroscopic, and it doesn't. You can superpose electron states, but not cats. The jargon term here is "decoherence." The cat contains so many wavicles that they all get tangled up together and ruin the superposition in less time that light can travel the diameter of an electron. None the less, on suitably small scales-and we are talking very small stuff here, not anything you can see in a normal microscope-the universe behaves just like quantum physics says it does.
One final ingredient will set you up for the rest of this guide: general relativity. This is what emerges from ordinary relativity when you throw in gravity. Einstein found that space and time are part of one thing and that that thing is bent. Moving objects "feel" the bend as a force and that force is gravity. This theory again works very well, explaining the origin of the universe in the big bang, its current expansion, and several other cosmological puzzles. Not all, but that's another story.
Vibrating loops in different dimensions
So far, so good: two theories, one exquisitely accurate on the scale of the entire universe, the other on the scale of an electron. But, as we have seen, where they overlap the theories contradict each other. This upsets the monotheistic tendencies of physicists. They can't both be true, and possibly neither is.
This is where strings come in. And superstrings, hot on their heels. What physicists have been seeking, ever since Einstein, is a Grand Unified Theory of physics-a GUT. Or, in slightly self-deprecatory terms, a Theory of Everything-a TOE. This is a theory that agrees with relativity on large scales, with quantum theory on small ones, and holds together in the middle. And some of them think they've found it.
Something has to be changed, of course, or nothing works. In string theory, what gets changed is the nature of those fundamental wavicles out of which everything is made. The basic idea is that relativity is about space and time, which are extended quantities-they are curved, they have a shape. Quantum is about wavicles, which are point-like-they have position but no shape. So the first step is to replace the points with little loops, or surfaces, or something more exotic that can have a shape. These are called strings.
The great thing about little loops is that they can vibrate, like a violin string. Each vibrational pattern corresponds to a quantum state. The number of waves that fit together round the loop has to be a whole number-1, 2, 3, 4, whatever. On a violin, these different patterns are the fundamental note and its higher harmonics. So quantum becomes a kind of music of the spheres-well, music of the strings. It was all fitting together very neatly indeed until it turned out that space-time can't be four-dimensional if string theory is to work. For a while the favoured number of dimensions was 26, then 11, and nowadays it is mostly ten. Three of these are the traditional dimensions of space-north-south, east-west, up-down. A fourth is time. The remaining six are (perhaps) curled up very, very small, so you can't see them, and they are the dimensions of the little loop, or surface-the string.
Extra dimensions? What are those? About 120 years ago an English clergyman and schoolmaster Edwin Abbott Abbott (yes, two Abbotts) wrote a short book called Flatland. It was about the adventures of Mr Square, who lived in the two-dimensional space of a Euclidean plane. Mr Square, a sensible sort, did not believe in the absurd notion of the third dimension. Until, one fateful day, the sphere intersected with his plane and flung him into realms he could never have imagined.
Flatland was a parable about the fourth dimension, a topic of considerable interest to the Victorians for scientific, mathematical and theological reasons. To the hyperspace theologians, the fourth dimension was an excellent place for God to sit, outside his creation but in intimate contact with every point of it. To the scientists, time could profitably be thought of as a fourth dimension: north-south, east-west, up-down, past-future. To the mathematicians, new dimensions were springing up in every problem they studied: they were just extra co-ordinates in a conceptual space.
In everyday parlance, we usually think of a dimension as a direction. Space has three dimensions because there are three independent directions: north-south, east-west, and up-down. Using these directions, we can locate every point in space (exactly three locations are needed-six miles north, five west, and two up, say-relative to some chosen reference point). Space-time involves a fourth "direction," past-future. So we feel comfortable with Einstein's idea that the space-time continuum is four-dimensional.
Mathematicians and physicists have a more relaxed view of what a dimension is. To them, it is just an independent quantity that can be represented by a number. Think of the moon moving in space. In order to specify its "state"-where it is and what it is doing-we need six numbers: three to locate its position in space, and three more to tell us how fast it's moving. The conceptual "space" in which a mathematician's moon moves has six dimensions, not three. The extra three really are needed, otherwise there's no sensible way to describe the moon's velocity, which is important-among other things, it is what keeps the moon up there instead of crashing to earth. The solar system, with its nine planets and single sun, is a 60-dimensional space-six numbers for each of the ten bodies. Add the moon, and it's 66. Include the other moons, and we're talking hundreds.
The ten dimensions (or 11, or 26, or whatever next month's fashion may be) of string theory are co-ordinates of that kind. They are extra dimensions that specify the quantum state of what would otherwise be a featureless point particle. In string theory, every apparent point of space is really a tiny vibrating six-dimensional hypersurface-possibly a sphere, possibly a more intricate and surprising shape. Indeed it is in the interplay between the shape of the hypersurface and the usual four dimensions of space and time that a unified physics lies. The need to agree pretty closely with conventional relativity and quantum mechanics imposes constraints on the numerology. The details of how the physics is assumed to operate changes the implications of those constraints, which is why the answer might be ten, 11, 26, or even 1,066. Let's plump for ten.
Selectrons, supersymmetry and branes
If space-time is really ten-dimensional, why does it look four-dimensional? Where have the other six dimensions gone? After all, we humans exist in space-time. Why can't we perceive the stuff that we're made of?
There are two main types of answer. In both, we are unable to perceive the missing six dimensions, just as we don't see ultra-violet light (although we feel it with our skins if we sit out in it for too long) or feel magnetic fields. But we can fail to perceive those extra dimensions for two distinct reasons. The original string-theoretic explanation is that those extra six dimensions are curled up so tightly that we don't observe them. That vibrating hypersurface is so tiny that the most powerful microscope could never see it; it is as small as the fundamental units of space and time (possibly because it is a fundamental unit of space and time). From a distance, a length of spaghetti looks like a one-dimensional object. From close up, however, we see that the spaghetti also has thickness, and indeed thickness in two independent directions. At right angles to the length dimension we find a cross-section, which is a small disc.
Strings are like that. The cross-sectional disc is much smaller than it is for spaghetti, and it has six dimensions instead of two. But those are technical details. Indeed, by adding even more new dimensions we get superstrings, which bring an elegant unity to a motley crew of fundamental particles. A superstring viewed from one direction looks like an electron. From a different direction it looks like a selectron, the supersymmetric partner of the electron. Supersymmetry is a magic mirror, and everything in what we imagine to be the real world has its ghostly, inscrutable mirror image. We don't actually have any direct evidence that selectrons exist-but what the hell, the maths is pretty.
A newer alternative to strings is the brane theory. For about a century mathematicians have used the word "manifold" for a many-dimensional hypersurface, but physicists like to invent their own terminology to avoid giving credit where it is due. So they renamed m-dimensional manifolds "m-branes," a pun on membrane. The term "m-brane" is another name for an m-dimensional space, along with some assumptions about the laws of physics in that space. The plane is a 2-brane, ordinary space is a 3-brane. But a new term reminds us of different features, and here the important one is that branes of a given dimension fit neatly inside branes of any larger dimension.
Mr Square's beloved Flatland is a 2-brane immersed in a 3-brane. The sphere lives in the 3-brane, and can pass through the 2-brane at will. Mr Square has less freedom; the laws of physics constrain him to remain forever confined to his limited 2-brane, because he has no way to push himself in any direction other than those that lie in the 2-brane. Try pushing yourself into the past or out of normal space into the fifth dimension.
In this braneworld, the third dimension is not "curled up tight." It sticks out at right angles to the 2-brane, and it is vast. The sphere can move along it, and there is no reason in principle to limit the distance it can travel. Mr Square cannot perceive the third dimension because he cannot look in that direction, not because it is too small to see: he is blinkered by the laws of Flatland physics. His eyes are Flatland eyes, confined to the plane, and point only in the directions that Flatland occupies.
Similarly, our space-time might be a 4-brane immersed in a surrounding m-brane for some large m. Because we inhabit that 4-brane, none of our sensory or experimental apparatus can point outside our 4-brane world. If there's anything else out there, we have to infer it, not observe it. By the way, the favoured value for m is 11, not ten, because it is no longer necessary to make the extra dimensions small. The mathematics now works best in 11 dimensions.
Back to our leaky universe. If the universe is expanding, then gravity isn't holding it together as well as it should. One possible reason goes back to the early days of relativity, when Einstein wanted to get rid of one annoying term in his equations for gravity, but couldn't justify doing so. He called this term the "cosmological constant," and spent much of his life regretting leaving it in. One school of thought now considers this decision his greatest triumph: the cosmological constant implies that gravity becomes weaker than expected at large distances, and that would make the expansion of the universe accelerate.
However, that's a conventional relativistic explanation, not in the prevailing spirit of TOEs and GUTs and superdupersymmetry. Dvali and Turner's calculations show that the astronomical observations make perfect sense if instead of gravity getting weaker, some of it is disappearing altogether. When the sphere departed from Flatland along the third dimension, what Mr Square saw was a circle that shrank and vanished. If the universe of our perceptions is a thin skin of congealed four-dimensional milk floating on 11-dimensional coffee, then things might be able to leak out of the milk into the coffee, where we milk-bound entities can no longer perceive it. So where has our missing gravity gone? Seeping out into those extra dimensions. There's a hole in my universe, dear Liza, dear Liza...
