26.02.2009 21:47
Compactified spaces
As you probabely know the idea of there being more than 4 dimensions is a very persistent one in speculative physics. Most prominently there is string theory which needs 11 dimensions, but there are many other fancyful ideas. (Which are a bit more economical in the number of dimensions they need). Me myself I work on such a model for my PhD.
Normally, the dimensions d > 4 are discussed away from every day experience by claiming that they are compact and small and hence we cannot walk into them. They only unfold for a very high energetic probe.
At least in popular books these ideas are illustrated with a picture like this:
That is people tell you that you should imagine a small circle or (something more complicated as in the picture) attached at each point in your 4d space. That is you think of some lattice like in the picture, attach your circles and then take a continuum limit, ie let the circles move closer and closer together. Not so difficult is it? But this is blatantly wrong. Because according to this picture you have a compactified space at each point on the plane, but once you move into this plane, you get detached from the surface holding it. Whereas in reality at each point in the compact manifold there also extends the extended plane again. And from there it continues until you get mad.
That is to say the above picture is very different from the one of a infinite cylinder of radius R:
In the cylinder, which is extended in 1 dimension and compact along another, when you move along the circle at every point you have a inifinite line. Staying whith this two dimensions the above picture would correspond to a straight inifinite line, where there are circles attached at each point. So at every point in the line you are free to move into the circle or to continue along the line, but once you have left the line and entered a circle, there is really only one direction open to you until you have walked all along the circle. Or take a torus (donut) which is 2 dimensional and compact in both directions. Each point is the intersection of two circles. The thing above isn't such.
Honestly, I can imagine 3 dimensions because that's the space I live in, and then I am fine with the fourth, because I know what time is, but all the other things I have no intuitive grasp.
J'ai pris l'image de ce site.Merci
