JOURNEY TO THE 10TH DEMINSION
Physics can't find the biggest thing in the known universe, so it's looking beyond our paltry three dimensions. Michael Moyer enters the zone of insanely hard mathematics, translates what he finds into plain English, and makes it back alive.
by Michael Moyer
March 2004
di · men · sion n1: a measure in one direction 2: the number of variables needed to locate a particle 3: a property of space, or the space-time continuum, related to extension in a direction.
If the following seems ridiculous, far-fetched or just outright outlandish to you, rest assured: It is. It will probably hurt your brain, as it has hurt mine, and as it most definitely hurts the brains of those who come up with this stuff for a living. The following asks you to accept ideas that are counter to the fundamental basis of our experience, the framework through which we comprehend everything from setting down a coffee cup to the arc of a home run as it sails into the upper deck. The basic point of what follows -- and by the way, what follows is not fanciful provocation but has been worked into contemporary consciousness by the brainiest physicists alive today -- is that everything that you have ever experienced has in some small but significant way been an illusion. Why? Because everything you have ever experienced you have understood as happening in three dimensions of space -- up-down, left-right and front-back. Yet this is not how things happen. Things happen in more than three dimensions of space; to see them in only three is to succumb to a trick that the universe is constantly playing on us.
Space as you know it is a lie. What follows is an approximation of the truth, or at least of various conceptions of the truth. There is no one model of the extra dimensions in the universe, no one statement of fact that all physicists can agree on and create in their computers. Alas, things are not that simple. There are at least two and possibly three completely different theories of what these extra dimensions should look like. And in each of these theories, the specific form of the extra dimensions -- their shape, whether it be Gehry-esque or nail-straight -- is unknown. But let's not let that intimidate us. Let's get started.
Type of possible space #1 : A 10-dimensional universe made up of the normal three dimensions of space, plus one of time, plus six-dimensional Calabi-Yau manifolds located at every point in normal three-dimensional space.
Excellent question #1: What the hell is a Calabi-Yau manifold?
Attempted answer #1: It is arguably impossible to imagine what a Calabi-Yau manifold is, because it has six dimensions. But let's try anyway. A Calabi-Yau manifold looks kind of like a balled-up piece of paper, except it's one whose curves and twists and turns are intricate and Möbius-like, looping back over and around themselves with clear disdain for Euclidean geometry. A Calabi-Yau manifold knows no straight lines. I try to imagine myself inside one of these manifolds: I think it's probably much like a fun house, mirrors everywhere deflecting your gaze in every which direction, so that at any time you could be looking straight ahead and see, for instance, your back. Except it's not quite that -- there are no mirrors in a Calabi-Yau manifold, there is only space itself. So while you can still look forward and see your back, you could also, theoretically, throw a baseball at it, only to feel the little missile smacking your spine two seconds later. That baseball might have traveled up and around, roller-coaster-like, through six dimensions, eventually ending up at your back. A Calabi-Yau manifold is a strange thing indeed.
To reiterate: I'm not making this up. I am only attempting to report to you, dear reader, what I have heard smart people say, and what I have read in scientific papers and heard at conferences, and to report it in a way that you and I might be able to get our heads around it all. My attempts will necessarily be futile and inaccurate, because I write in English.
When scientists talk about extra dimensions, they actively avoid the use of English, tied as it is to our everyday experience of space and time and reality. English is by its very nature misleading, imprecise. So they use the language of math, whose concepts and terms are easily generalized into any number of dimensions or spaces or inconceivable, unphysical situations.
Consider how mathematicians think about the difference between a circle and a sphere. To a mathematician, a sphere and a circle are essentially the same thing, a collection of all the points that lie equidistant from a single point. (Think about this for a moment: If you took a piece of paper, then marked a point on the piece of paper with a dot, then marked all those places on the paper that are exactly, say, 1 inch away from the dot, you'd have a circle. Same thing with a sphere, but you'd have to mark all the points in three dimensions.) Mathematicians call circles 1-spheres, as creating them requires only a one-dimensional line, properly curved. Mathematicians call actual spheres 2-spheres, as creating them requires a two-dimensional surface. To mathematicians, the distinction between a 1-sphere and a 2-sphere is insignificant. They prefer to study n-spheres, spheres that can have any number of dimensions you like. No matter that we cannot imagine what even a 3-sphere would look like, sitting as it would in four-dimensional space. No matter that we cannot describe its appearance in English, or Japanese or Latin. Mathematics describes it with precision, and mathematics is the only language that counts. ( In general, an n-sphere of radius 1 is described by the equation { x ∈ Rn+1 | d(x,0) = 1} )
I learned this when I took a graduate-level mathematics course from Brian Greene, the Columbia University physicist who has done a very nice job popularizing string theory, the theory that requires our universe to be made of 10 dimensions. (Actually, recent developments in string theory suggest that there may be yet another dimension, for a total of 11, and that this new extra dimension is invisible because it is "curled up" into an infinite number of tiny loops. But to avoid further brain pain, let's stick with 10.) At the time I took the class, I was working toward a master's degree in the philosophical foundations of physics. This course was by far the most difficult one I have ever taken. After Week 3, I understood very little of what was going on. Yet the dimension stuff, that was Week 2, I think. Anyway, the course was this year-long journey through the world of differential geometry, which, as far as I could tell, should have been named abstract geometry, concerned as it was with the properties of surfaces and spaces of things in ndimensions. (Remember, nhere can be any whole number you wish -- 2, 5 or 12,497.) The culmination of this course, which paused only once for a quiet moment of repose somewhere around Week 8 when Professor Greene revealed to us that we had just derived the fundamental equation of general relativity (who knew?!), was an introduction to the basics of string theory.
Now, recently I've been busy with the day-to-day of magazine work. My string theory, if I could ever claim to have had any string theory, is a bit rusty. But when I dipped back in, I was lucky to find a Virgil to guide me through the various levels of theoretical-physics hell, someone informative and protective who understands both my fascination and my confusion with the whole enterprise. His name is Subdoh Patil (call him Sub), and he's a graduate student studying string theory at Brown University. He invited me to what I understand to be the eighth circle of hell, full of the astrologers and the diviners, otherwise known as the Second Northeast String Cosmology Workshop. Here, in a lecture room at Columbia University, wise men spoke of the ways that string theory and cosmology -- the study of the universe -- may intersect. While I learned much from Sub's break-time translations of what was going on, I was heartened to find that he himself occasionally didn't get it. The field is too broad, too rich for any one person to grasp it all.
And why wouldn't it be? The reason we were at a workshop on string cosmology was that string theory carries with it great hope for both particle physics -- the study of the very small -- and cosmology. Both fields are beset with problems, "problems" here meaning deep chasms of ignorance in our understanding of the physical world. Both fields have been challenged by recent discoveries we don't understand. And both fields hope that string theory -- which explicitly requires the existence of 10 dimensions for the math to work -- will provide a way out of this mess.
The big problem : It's not that modern physics doesn't work, or isn't true or accurate. In just about every case anyone could conceive, the causal and statistical explanations provided by physics are bulletproof. Physics predicts that time should occasionally slow down? We do the experiment and find that, lo, time does slow down, and by just the right amount. Physics tells us that distant particles can instantaneously affect one another? Nearly fifty years later, we develop technology sufficiently advanced to take a look, and yes, particles behave in just that way. Modern physics is capable of revealing the intricate details of worlds whose existence we never would have suspected had not physics offered up strategies for observing and understanding them. Take, for example, the vacuum of deep space. According to modern particle physics, it is not empty at all. It swirls with innumerable subatomic particles constantly popping into and out of existence, eternally borrowing their short life span from the uncertainty embedded in quantum mechanics. And recent experiments have shown this new and unexpected reality to exist. Modern physics is the most powerful tool for understanding the universe ever conceived.
And yet it is wrong .
Well, that may be overstating it a bit. Theoretical physics is wrong in a few seemingly minor, carefully selected cases. Something called the magnetic moment of the muon (don't ask) is off by 0.00005 percent. And the predicted and experimental values of another thing, called sin 2θW (pronounced "sine squared theta W," though again, not really worth the effort), differ by 1 percent.
Oh, and there is another anomaly, one that is not so little. In fact, in terms of total energy, this thing, whatever it is, is the largest thing in the universe, about 14 times more energetic than the combined energies of all the stars and galaxies and black holes and protons and electrons and everything else we've ever found or thought was out there. In time it could grow strong enough to rip apart all the basic constituents of matter in the universe. We have no idea what it is.
It's not that people haven't taken stabs at determining what this stuff -- most popularly called dark energy, as it is energetic and mysterious -- could be. Yet these stabs are ludicrously wrong. How wrong?
Let Tbe the theoretical magnitude of dark energy.
Let Ebe the experimental value of dark energy.
If the theorists were right, then Tshould = E.
Yet T≠E.
T=Ex 10 120 .
(10 120 =1,000,000,000,000,
000,000,000,000,
000,000,000,000,
000,000,000,000,
000,000,000,000,
000,000,000,000,
000,000,000,000,
000,000,000,000,
000,000,000,000,
000,000,000,000)
There is really no good metaphor to help us grasp this degree of absolute wrongness. One may search far and wide for even semicommon objects and experiences that differ by 120 orders of magnitude. One will fail. It is greater than the difference between the volume of a drop of water and the volume of all Earth's oceans, innumerable billions of times over. It is greater than the difference between the size of a proton and the size of the observable universe. It's so wrong that it makes one think there must be a stupid error in the calculation somewhere -- someone forgot to carry a 1 and now we're left with this ridiculous result. But no, many have checked the math (though I haven't personally done so), and the embarrassing difference between what our theories predict and how the universe actually behaves obstinately remains.
(Actually, physicists prefer to call their theories "incomplete," rather than "wrong," as they are pretty sure that these theories can adequately describe and explain 99.999999 percent of everything in the universe. There's just that teensy little 0.000001 percent that can't be explained. Yet when smart people are trying to fix this "missing" 0.000001 percent by inventing completely new structures of space and time, telling us that we live in a universe with six or four or two extra dimensions that are out there but that we just can't see , that the universe may not be made out of the stuff that we thought it was made of but out of little strings instead, that there's some mystical Borgian antigravitational force out there that's more powerful than everything in the entire known universe 14 times over , then it seems to me, dear reader, that these problems are more pernicious than anyone is letting on. That is not tacking an extra bedroom onto an otherwise sound house. That is razing the foundation and starting all over again, possibly in another state where the tax laws are better.)
But if current physical theories don't cut it, and string theory still sounds a bit insane, perhaps we should try something completely different.
Type of possible space #2 : The universe as we know it is merely a three-dimensional brane suspended in a four-dimensional bulk.
Excellent question #2 : What the hell is a brane?
Attempted answer #2 : Here is the Powerpoint version of everything you need to know about branes:
· You live on a brane.
· A brane is like a membrane.
· Imagine the skin that forms on your soup when it gets cold. A brane is like that.
· A brane is some sort of lower-dimensional thing (the 2-D skin) sitting in a higher-dimensional space (your 3-D soup).
· Brane theory says our 3-D world is really just a brane.
· Our brane sits in a 4-D space called the bulk.
· Like so much congealed fat, we are prevented from escaping the brane and going into the higher-dimensional soup.
· Only gravity is allowed to do that.
This theory -- generally referred to as brane theory -- was devised in 1998, which is pretty recent by theoretical-physics standards (people have been noodling with various forms of string theory for about 30 years now). Three guys get the credit for coming up with it, one of whom -- Nima Arkani-Hamed -- was a 25-year-old fresh out of Berkeley with a Ph.D. He is now a professor in the physics department at Harvard. Smart guy. He and his cohorts Savas Dimopoulos and Gia Dvali had been working on a problem that had confounded big thinkers for, oh, a few decades, when they suddenly realized that all they had to do to get us out of it was to invent another dimension! Voilà!
The problem that had been confounding all of these smart people for so long (and continues to confound them; did I mention that none of what I'm describing has yet been supported by a shred of experimental evidence?) was this: Gravity is weak.
Objection #1 : That's silly. Gravity is the strongest thing around -- it's what moves planets and clusters of thousands of galaxies, not to mention that it's what keeps us pinned to the ground.
Rebuttal #1 : When you compare it to the other forces -- say, the electromagnetic force -- gravity is incommensurably less powerful. Take for example a simple refrigerator magnet. Think about the forces acting on it as it pins a photo to the fridge. There's the combined gravitational force of the entire Earth pulling the magnet down to the ground, and the magnetic attraction of a little strip of iron anchoring it to the fridge. Those few grams of magnetic material win; not even a planet-size helping of gravity is enough to overcome its intrinsic weakness.
Objection #2 : OK, so gravity is weak. But that's just the way it is; physicists can't do anything to change gravity's strength. All they can do -- all they're supposed to do -- is describe it.
Rebuttal #2 : Correct, sort of. There can be many ways to accurately describe something in nature. Yet there is only one way in which the thing in nature actually works, one physical process that determines how things happen, one Truth (big T) of the universe. Right now, particle physicists have a way to describe the workings of gravity. And while they think this description is useful -- it accurately predicts the outcomes of experiments and the like -- they do not think that it reflects the true physical processes that govern the universe.
The world according to the current theory of particle physics seems very ad hoc, in that it must treat the weakness of gravity with great care, it must introduce new assumptions and fine-tune all the parameters it can in order to replicate the weakness of gravity. Everything else works fine; gravity is the oddball of the particle family.
Unfortunately, to go any further -- to describe exactly how modern particle physics treats gravity, and henceforth the difficulty of coming up with a reason for why it should be so much different from the other forces, requires a little refresher course on the state of particle physics today. The current model, which has become so well tested and generally accepted that everyone just refers to it as the standard model, was the major accomplishment of physics in the second half of the 20th century. And everyone believes it is accurate, though no one believes it is True, and the person to replace it will probably be the 21st century's Einstein.
Unfortunate but essential aside summarizing the present state of particle physics:
The standard model describes how everything in the subatomic world works. It is the ultimate (for now) and most general (again, for now) extension of quantum mechanics. It is basically a listing of all the fundamental particles and a set of rules governing how those particles interact. And how do they interact?
Particles interact by exchanging particles with other particles. For example, an electron exerts a force on another electron by shooting a little photon (a particle of light) out to the other electron, which the second electron catches and responds to. The preferred anthropomorphism is that particles "communicate" forces using "mediating" particles, like photons. This is what the process looks like in my head:
As you can clearly see, the two electrons "communicate" by tossing the basketball-like photon back and forth to each other. This tossing pushes the electrons apart, which agrees with what we see in the world -- negatively charged electrons repel one other. With particles other than electrons, the net effect can be attraction, not repulsion, but the principle remains the same. This is the essential point necessary to understand the rest of this stuff: A force -- any force -- is caused by things throwing particles at other things. The more particles that are thrown (and caught), the stronger the force will be.
OK, so where does gravity fit into all this? Just treat it like any other force -- gravity is caused by massive particles throwing "gravitons," attractive particles, at each other. These gravitons work to pull massive particles closer together. Simple as that. (Aside within the aside: You may have caught wind of another theory of gravity called general relativity. A fellow named Einstein came up with it almost 100 years ago. Conceptually, it could not be any more different from the standard model. General relativity explains gravity by invoking the warping of space-time; the standard model explains it and everything else by invoking the exchange of subatomic particles. Problems happen when we try to put the two theories together, when we try to describe things that are both very massive and very small, like black holes. Problems like mathematical inconsistencies, zeroes in denominators, nonsensical results. String theory has been developed at least in part to avoid these problems and combine quantum mechanics with general relativity, using a new structure of space-time and all that stuff I talked about a few pages ago.)
End of aside .
Now it is finally possible to understand why gravity's weakness is such a pressing problem. Within the standard model, there is a symmetry between the graviton and the other force-carrying particles. They share a common conceptual description. This common description implies that the forces the particles produce should also be similar, in both character (for example, the theory correctly predicts that the strength of both the electromagnetic and gravitational force diminishes with distance) and in magnitude. Yet, as we have seen, gravity is much weaker than every other force. And so we are left with the question: What makes gravity so special?
Enter brane theory. Recall that brane theory postulated that we are trapped in our three-dimensional world, which is itself floating in a higher-dimensional space. We cannot travel into this higher-dimensional space. In fact, nothing we know of can travel into it -- not electrons, or quarks, or exotic muons -- except for the graviton. It alone can journey into the higher dimension. And as gravitons spread out into that extra dimension, there are fewer here to do the work of pulling heavy things together. As we learned earlier, a force -- any force -- is the result of particles throwing particles at other particles. When there are fewer particles being caught, that force gets weaker. According to brane theory, we lose gravitons out into the fourth dimension. The result: Gravity is weak.
Please imagine that you are a graviton. I admit, I haven't given you much to go on about what it would be like to be the carrier of the gravitational force. This is because I have no idea what it would be like, either. Gravitons, like photons, do not possess the property known as mass. They weigh nothing. Because of this, they travel at the speed of light. And according to Einstein's other theory of relativity (the special one), anything traveling at the speed of light does not experience the sensation of the passage of time. I find this difficult to imagine, as I am imbued with the quality of mass, and thus ineligible, according to Einstein, to travel at the speed of light. But bear with me.
So you are this graviton moving along through the universe at the speed of light, except the universe, to you, is bigger than it seems to a nongravitational particle. Not wider or higher or longer, but fundamentally bigger. In addition to the three directions your fellow particles can move in, you can move in another direction, in the direction of the fourth dimension.
Now, there is some discussion about just how large this extra dimension has to be, and what it must look like. Here are a couple of options:
1) The extra dimension is small and round, though not nearly as small as the truly minuscule dimensions associated with string theory. It can be up to about 1 millimeter long. One millimeter in particle physics is to us like the distance between here and the nearest quasar. It is magnificently large. For this reason, brane theory was originally called the theory of large extra dimensions (the theory, like our sphere earlier, is general, and can describe any number of extra dimensions you wish). Each extra dimension is curled up in a circle, so if you were a graviton, you would be able to move in this circular direction at every point in space, while still moving in a straight line through the three dimensions we all know.
2) The extra dimension is infinite. We are all very familiar with infinite dimensions. In fact, the regular dimensions we move around in every day are infinite. Theoretically, we could choose a direction -- let's say "up" -- and move in that direction forever, never coming back to the same place, never reaching an end. When Lisa Randall and Raman Sundrum -- she of Harvard, he of Johns Hopkins -- first proposed the possibility of an infinite fourth dimension back in 1999, it was the first time anyone had taken seriously the possibility that an extra dimension doesn't have to be tiny to be invisible. If only gravitons can escape into the extra dimension, then it could be perfectly straight and infinite and we'd never know. This proposal has the advantage of an extra dimension that looks just like the dimensions we know very well. Except that only gravity experiences it.
There's a third option, but it's fundamentally different from the two options above. Notice how those two require only one extra dimension. (Some variations of these theories feature two or three, but they only require one.) This is because gravity needs only enough extra space-time fabric to dissipate and grow weak. Whatever intricate particulate structure you wish to lay on top of this fabric -- whether it be string theory or another option called supersymmetry or something else entirely -- is just gravy.
But this third option is more of a combination of string and brane theory, a way to integrate the major ideas in one big package. In this scenario, there are two extra dimensions, both of them small and straight and finite, which lead us to a parallel universe.
Hear me out on this one. Remember how string theory required us to live in a 10-dimensional universe? Well, imagine that instead of living in a four- dimensional space (three plus one of time) and having the extra six dimensions curled up into balls so small we have no hope of ever directly observing them, there are two four-dimensional spaces: ours and a different one hanging out not too far away from ours. A two-dimensional surface connects these two branes (remember, a brane is the thing we are trapped on), which gives you 4 + 4 + 2 = 10 dimensions. This implies that there is another, mirror brane located as little as a millimeter or so away from us at all times, but which we can never reach, because we are not gravitons. This is the sort of thing that the string theorists come up with and think not "how manifestly ridiculous," but -- and I'm paraphrasing here -- "wicked awesome!" It does make the math work, however, even if it breaks our brains.
What I have been describing is really a trip, to put it mildly. These theories ride the very edge of human understanding. No one yet knows if any of them are true, because no one has figured out how to conclusively test them against the firm foundation of the physical world. They are still too ethereal, too vague, too ill-defined. But these theories will ultimately become clearer both to the people who are devising them and to us. The search is on for a True theory of the universe. And the way things are going, that theory will describe a universe that we will completely understand yet still not be able to imagine.
Michael Moyer is the senior associate editor of this magazine .
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