A reader asked, "How many dimensions are there?" But first, it is worth answering, "What is a dimension?"

Without defining it too rigorously, I would say a dimension is essentially a direction. If we consider a drawing on a sheet of paper, it has two dimensions. This is because there are two directions in which you can move your pencil. You can move left and right, or you can move up and down.* If you want to move your pencil from point A to point B, you have move left/right a certain distance, and then move up/down a certain distance. That is, it takes two numbers to specify the coordinates of any point. Two numbers, two dimensions.

*We don't count "up" and "down" as distinct dimensions, because if you move up by a negative distance, it's just as if you had moved down.

Of course, our universe is not a mere drawing on a piece of paper. In our universe, we have three dimensions. You need three numbers to specify the location of any object. Since we're all Earth-dwellers here, the most convenient set of numbers to use are called latitude, longitude, and altitude. That is, the directions are north/south, east/west, and up/down.

But I omitted a fourth dimension: time. I think you'll agree that time is rather different than the other three dimensions. But it is a dimension nonetheless. You may argue that it is impossible to move backwards in time, but that is not important. What's important is that if we compare event A and event B, we need three numbers to specify the relative location, and one number to specify how much later/earlier event B is. The difference between time and space is a big one (don't let any amount of physics talk convince you otherwise). For instance, it is easy to get the directions north, south, east, and west all confused, because there's hardly any difference when you rotate yourself around. But no matter what you do, you will never confuse time and space with each other. (That said, there is a bit of space-time "rotation" that occurs when you approach the speed of light.)

Therefore, four dimensions is my final answer.

Wait, you wanted to hear about String Theory? Okay. It used to be that various versions of String Theory posited that the universe has 10 dimensions, or 26 dimensions. But in the 1990s, the science progressed, and the current most accepted version, called M-theory, now posits that the universe has 11 dimensions. This often prompts questions like, "WTF are those physicists thinking?" If I understand correctly, they are thinking that String theory is the most promising way to solve the problem of quantum gravity, because it naturally predicts the existence of a spin-two boson which behaves like the graviton. But that answer's much too arcane for my purposes here, so don't worry about it.

But it is a good question to ask, "How could the universe possibly have 11 dimensions when we so clearly see 4?" It is not merely a philosophical question. There are some things in physics that absolutely rely on having only three space dimensions. In particular, there is the so-called inverse-square law. If you have a light bulb in the middle of the room, then its light spreads out in all directions. The further you are away from the light bulb, the more the light spreads out before it gets to you. Similarly, the further away you are from the Earth, the more its gravity spreads out. The more spread out it is, the weaker its strength. More precisely, the strength of gravity is proportional to the inverse-square of distance. This is directly related to the fact that we have three dimensions of space. For instance, if we had four dimensions of space, we would instead have the "inverse-cube" law. It's not just humans who can't access the extra dimensions implied by String theory--it seems that gravity and light can't access them either!

This post got split into two. See part two: Dimensions in String Theory

## Monday, February 2, 2009

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## 6 comments:

I don't have a problem in general with the idea of eleven (or more) dimensions in string theory--as far as I understand it, string theory is a model that allows us to make predictions about the universe, just like any scientific hypothesis or theory. If it takes eleven dimensions in order to produce those predictions, fine--they don't all have to correspond to space-time in an obvious fashion.

This could be a fun series of posts--there are a number of mathematical concepts of dimension that each capture some essential properties of what we think of as "dimension": dimension of a vector space, dimension of a manifold, fractal dimension, probably more.

With regard to your discussion of the dimension of a piece of paper, I think a point ought to be clarified: there are of course infinitely many directions in which you can move a pencil. The important thing is that once you've fixed two directions, any other direction you can move is a combination of these two.

It's only a two-part series. :-) And I don't think I'll really get into vector spaces, or indeed anything that is mathematically rigorous. But maybe in the future I'll write something about fractal dimensions.

I've always considered that any spatial dimension above 3 would be external to the material of our universe, although it would affect our experience of it (eg a mass causing a beam of light to bend).

The classic rubber sheet is a good way to visualise this is. Put a metal disk on that sheet and it will distort the sheet downwards. An ant that lived in this universe would only see 2 dimensions but would feel a force drawing it toward the disk. As beings external to this universe, we see that it is explained by including a third dimension, but also see that this extra dimension wouldn't have any effect the inverse relationship between the perceived brightness of a light source and the distance from the light source in this 2D universe.

I think that there are 10 dimensions but only 5 nickelensions ;-)

I suppose you could simply say that the extra dimensions in string theory are simply a mathematical abstraction, that they are not really "real" in the sense of being part of the material universe. But put it this way: if you think of the extra dimensions as real, maybe you will find an explanation for why we can't access them. And maybe that explanation will predict that certain kinds of particles

canaccess the extra dimensions.I guess I'll get into a bit of this in the next post.

I think it would be an incorrect supposition that gravity "can't access" the other dimensions. In fact, it's the existence of the extra dimensions that may account for the relative weakness of gravity when compared to the three other fundamental forces.

It is postulated that perhaps gravity's strength is being diluted as it works through all eleven dimensions that may account for that discrepancy.

You are correct. Many theorists are excited about the possibility that extra dimensions will somehow explain the relative weakness of gravity compared to the other forces.

My point here is only that it

seemslike gravity is limited to three dimensions, because it follows the inverse square law. Therefore, the simplest 11-dimensional model would not suffice.Post a Comment