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V. Extra Dimensions

Space is three

As we indicated before, one of the key consequences of string theory is that there are more dimensions to our world than we imagined. We normally think of our world as four-dimensional. The count goes as follows. We can think of a point in space as being specified by its left-right position, its height, and its depth. Space is therefore three-dimensional. We need three numbers to specify a point in space. We can redo the count in another way. Consider the surface of the earth. When we specify the parallel and great circle of a position on the globe, we can pinpoint the position. The surface of the earth is two-dimensional. But if we had buried a treasure under its surface, we would need to know also how deep it is buried to locate it precisely. Space is again three-dimensional according to this count. A similar count is of course valid for the localization of stars. (Exercise: Count !)

A special dimension

Time is the fourth dimension. Indeed, to localize an event, we not only have to specify its precise position in space, but we also need to know when it happened. The extra number we need is the time at which the event happened. That fourth number indicates that space-time is actually four-dimensional. Recall that it was one of the big achievements of special relativity to treat the three dimensions of space and the one dimension of time in the same mathematical framework.

Of course, the time dimension still plays a special role, and its role in string theory is similar to its role in our everyday lives, or in the theory of special relativity. The extra dimensions we consider in the following are of the usual spatial sort. -- There are theories that try to make sense of two different times, but we do not consider them here, since they have little or nothing to do with string theory. (Note: crackpots tend to underestimate how seriously professionals have already investigated the ideas they come up with.) We concentrate on extra dimensions in space.

More than 3+1

Many scientists had played around with extra dimensions before string theory came along. The idea is natural, since an extra dimension gives some room to play around in, and circumvent theorems that tell you that something is impossible in the 3+1 dimensions that we know off. But people had considered only one extra dimension, since they didn't need more than one. String theory actually tells us there is more than one extra direction. How many more ? Now, that's a tricky question. For years we have thought that string theory needs precisely six extra spatial dimensions -- we will assume this to be true for now, and will explain some more subtly points about how to count the extra dimensions later, when we introduce the concept of M-theory.

9+1 = 10

String theory tells us we live in ten dimensions. How does it tell us that, and, importantly, why don't we need to specify ten coordinates when we want to specify the location of a treasure ? The first question is the more difficult one. The mathematics of string theory is such that it leaves us with a dilemma. We either choose to have ten dimensions, or we can choose to accept that there are particles that have a negative probability to be in the universe. The last option (and its formulation in the last sentence) is entirely nonsensical. In other words, nobody has made sense of the notion of "negative probability" up until now, and it is doubtful whether anybody ever will. We now what it means to have a particle somewhere with a small or high likelihood, but not what it means to have it exist with a "negative" likelihood We choose therefore for the lesser evil, and interpret the conundrum as the fact that string theory simply predicts that there are ten dimensions. The problem actually turns out to be a blessing in disguise. We get one more spectacular prediction from string theory.

Little balls everywhere

Let's tackle the second question then: where are the six extra dimensions that string theory predicts ? There are different answers to this question, and for starters I discuss only the old one -- I'll come back to an exciting new possibility later. -- The first part of the first answer is that the six extra dimensions are very small, or compact.

They do not extend very far, in contrast to the three spatial directions that we know of. The extra dimensions are curled up, and in such a way that they are extremely tiny. Since we haven't seen them yet in particle accelerators, we know that they are smaller than 0,000000000000001 meter. The second part of the first answer is that these extra dimensions are everywhere. Indeed, we can think of every point in our space (or kitchen) as not actually being a point, but as being a tiny six-dimensional ball. We do not need to specify the six extra coordinates of a knife in our kitchen, say, because the ball is so tiny that we can easily locate the knife without this extra information. If these dimensions were bigger, we would have seen them long time ago, of course. We would have moreover been able to think much easier in 3D, 4D, or even 9D. (Note that the trick of hiding the extra dimensions is very similar in spirit to hiding the stringy features of strings -- both make use of the fact that the resolution of our measuring devices is too small to make out the new features, as yet.)


We first of all saw that there is no real problem to having six extra dimensions, as predicted by string theory. That is not to say that they do not have observable consequences. Indeed, once we probe small enough distances, we should be able to see many new interesting phenomena, depending on the shape and size of these extra dimensions. For one thing, we will be able to distinguish particles that run around in these extra dimensions from the ones that don't. Moreover, we will be able to see particles that are actually strings or membranes that are wrapped around some directions of the six-dimensional compact space. And many more interesting phenomena would appear and they might be observed in the next accelerator, depending on how small the six-dimensional compact space is precisely. Let's hope it is not so small that we will never be able to see it in our lifetime.

Brane worlds

We treated some aspects of a first answer to the question of where these extra dimensions are hiding. There are other, more modern answers to this question, of which we will treat one in more detail in the chapter on branes. But to introduce branes properly, and to understand how they provide an alternative answer to that question, we first need to introduce a few more properties of string theory.


Illustrative footnote: An expert points out to me:

"Dear J,

I was wandering around the net, and encountered your web site on string theory. It looks very nice, a really thoughtful service.

One comment: I would quibble with your characterization that string theory predicts 10 dimensions. It is not known to predict
this--string theory can be formulated in any dimensionality; it is just that one does not have a classical solution with exactly flat space in any dimensionality. The leading effect of venturing away from the critical dimension is that one finds a tree level dilaton potential, which forces us to solve the dilaton and graviton equations of motion nontrivially rather than finding a constant-dilaton
Minkowski space solution. Other contributions to the dilaton potential from branes, fluxes, orientifolds, etc. allow us to find compensating forces on the dilaton that fix it. (Or one can consider the linear dilaton background in cases it makes sense,
as in the 2d string.) Since we don't live in precisely flat space, the critical dimension is not a condition we know to impose
even phenomenologically. It might transpire that SUSY provides a reason to focus on the critical dimension (I also find this a very plausible possibility), but this is not known at the moment either experimentally or theoretically. Best regards, E"

To translate: when I say string theory predicts 9+1 dimensions, my colleague believes I make quite a few hidden assumptions (that I may need to explain in lay terms). My colleague is right, of course, and she refers to established technical results to support her case. I include the comment as a footnote, not merely as a fair correction to the above, but also to illustrate that we discuss about how to present string theory results to the layman, and that when we argue, we can use a technical language not accessible to all. To truly do justice to her comment, the task set out for me is to explain the content of the comment (on which experts agree) in understandable terms. I may implement this later on ...