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31 Aug 2007

What are dimensions?

Filed under: Physics — paulcook @ 6:57 pm

Since I’ve hardly been filling this blog with posts recently, I thought I might post an email I wrote recently, in reply to a question I received about the nature of dimensions. Mine is by no means a complete answer, but maybe it’s interesting. Follow-up questions welcome!

The question:

I have been reading a lot of books and web sites on string theory. It all seems very interesting, all these extra dimension and so fourth. I was just curious, it is supposed that these extra dimensions could be real, I have yet to read how the first three dimensions that we take for granted in this universe are real physical things. I know that the term dimension is used in plotting locations and trajectories of objects in space on paper, but are they real physical things that exist in the real universe?
I would really appreciate your help in this.

That’s a tricky question. To start with the three dimensions of space (and one of time) that we’re well aware of: they’re real in that they are what makes space, well, space. The idea of “space” is that it provides somewhere that things can be — without dimensions, there’d be no way to talk about where something is, how far apart things are, and so forth. Motion is merely the movement of things within the space.

Now that’s all there was to space, before relativity. Einstein’s General Relativity shows that space itself is a “dynamical” object, which means essentially that it is something that can change. Basically, matter and space interact — space is how you define where something is, but the presence of something (matter and/or energy) in turns affects the lengths of nearby pieces of space. So light travelling near a star is “bent” by the gravity of the star, as a result of the mass of the star affecting the definition of coordinates and motion nearby. So in this respect dimensions and space become a physical entity on which matter has an effect.

The additional dimensions predicted by string theory are no different to the three (plus time) that we’re used to, at least conceptually. When specifying the position of something, we just need to specify locations in each of nine directions, as well as a time. However, the fact that in our daily lives we only experience three of those spacial dimensions means that the other six are somehow irrelevant on large length scales. This might because they’re “rolled up” really small, by which I mean that the possible range of positions in that dimension is very small, and so everything is so close to everything else in that direction that we can’t even tell that there is another dimension. Alternatively, the particles from which we’re made might be in some sense “trapped” on the surface of a three-dimension object in the nine spacial dimensions.

However, in a very real sense we still don’t actually know what spacetime really is. Quantum gravity considerations strongly suggest that space is not continuous on the smallest scales — there should exist a smallest possible length, the Planck length. Any length smaller than this makes no sense. Furthermore, gravity and the effect of mass on spacetime arise in the string theory in an exactly analogous way to that in which particles arise — as specific vibrational modes of strings. So this seems to complete our growing re-interpretation of spacetime from being merely a fixed measurement apparatus on which physics happens, to being itself a part of physics. This makes physics very difficult, however, as most of our current techniques rely on the existence of concepts like detectors at spacial infinity, or being able to define a universal starting time for an interaction. So in my opinion certainly one of the interesting areas that string theory will be exploring in coming years is “emergent geometry”, where concepts that look at large scales like spacetime will turn out to arise from quite different interactions in some theory that resembles string theory in certain regimes.