Dimensions in String Theory

See the previous post in this two-part series: Dimensions in the Universe

One of the obstacles in String Theory is that it predicts 11 dimensions (10 of space, 1 of time). Clearly, our universe has 4 dimensions (3 of space, 1 of time). So how could String Theory possibly be a good description of reality?

To reconcile this discrepancy, it is good to know why String Theory predicts so many dimensions. The basic idea of String Theory is that elementary particles are not point-like objects, but are tiny string-like objects. How tiny? If particles are point-like particles, then their size is precisely zero. But according to String Theory, they are strings with length of roughly 10^-35 meters. Let's just say that if you shrunk down to the size of an atomic nuclei, then you'd be less than halfway there. One of the problems with String Theory is that, due to the uncertainty principle, it is practically impossible to directly observe the size of the tiny particles.

In order for this idea to be consistent with the kinds of particles we see, the strings need to be able to vibrate in 10 different directions. To the string, the seven extra dimensions are just as real as the first three. But you have to remember that these strings are tiny! If the strings can see the extra dimensions, but we cannot, perhaps the extra dimensions are just really small.

What does that even mean, for a dimension to be small? There are essentially two different ways this can happen. Both of these will be illustrated using an analogy with a hose. (I'm stealing this analogy from Lisa Randall. But the graphics are mine.)

Let's say you're a giant. If you see a hose, you might think it was a one-dimensional object. You can only go up the hose or down the hose. If there is any multi-dimensional structure in the hose, it's just too small to see. The hose, as far as you're concerned is one-dimensional, the same way that a piece of paper is two-dimensional. Of course, paper has a small thickness, but for most practical purposes, you can ignore it.

However, if you were that little ladybug on the surface of the hose, then it would be a different story. Then you'd see two dimensions. You can go up and down the hose, but you can also go clockwise or counter-clockwise around the hose. Of course, if you go far enough clockwise, you'll end up back where you started. Nevertheless, it still counts as a second dimension. Strings, because they are so small, might be like that ladybug, with access to extra dimensions. But that's only one extra dimension. How do we get seven?
This is another way of drawing the hose, in more abstract form. The blue line represents the large dimension, that even the giant can see. The green loops represent the extra dimension that only the ladybug can see. But what if, instead of loops, we have more complicated shapes? For instance, if each loop were replaced by the surface of a sphere, we would get two extra dimensions rather than one. In String Theory, we replace the loop with a weird seven-dimensional manifold called the Calabi-Yau manifold. I have no idea why it needs to look that way, but that's what String theorists have found.

As I said before, there are two different ways to explain the extra dimensions. One is if strings are like that ladybug. The second is if strings are like the water inside the hose.

A hose is a three-dimensional object. If you are a giant, it appears as if the water can only go upstream or downstream, but in fact there are two extra directions in which the water can go. The water can also go up/down, and left/right. However, the water analogy is distinct from the ladybug analogy in an important way. The ladybug could go clockwise indefinitely, eventually reaching its starting point. The water cannot go left indefinitely, because there is a wall in the way. The walls of the hose confine the water to a small space, and that's why it appears to be one-dimensional to the giant.

In String Theory, there is something analogous to the walls of the hose. There are objects called membranes, or just "branes" for short. Branes can be of any dimension, and might be very large. These branes do not act quite like walls. Supposedly, each of the two loose ends of a string can be attached to a brane, unable to move away. This has prompted speculation among scientists that the entire universe as we know it is stuck on a very large three-dimensional brane. The three-dimensional brane would exist in a larger 11-dimensional braneworld.

To further complicate things, the graviton, which is the hypothetical particle which carries the gravitational force, is theorized to be a closed loop. It has no loose ends! Thus, gravity is unlikely to be affected by branes the same way that other forces are. If we're lucky, this fact can be used to explain why gravity is such a weak force.* Perhaps gravitons are escaping from our brane!

*Gravity may seem like the strongest, but think of it this way: the entire Earth is pulling you down, but it can be stopped in its tracks the moment your feet touch the ground. The electrons in your feet repel the electrons in the ground. The reason gravity seems dominant is because most electric charges cancel each other, while there is no "negative mass" to cancel gravity.

So you get the basic idea. There are a lot of exotic ideas that have been proposed to solve the dimension discrepancy in String Theory, but most of them are built on one of the two basic concepts above. Either the extra dimensions are made of tiny little loops, or there is a wall or brane or something which is preventing us from moving in the extra dimensions. The rest, as far as we non-String-theorists are concerned, are details.

[Most of this, I learned from Warped Passages by Lisa Randall and The Fabric of the Cosmos by Brian Greene]