On the train from Lausanne to Geneva Airport I recalled the way Polchinski introduces his chapter on toroidal compactification :

"In general relativity, the geometry of spacetime is dynamical. The three spatial dimensions we see are expanding and were once highly curved. It is a logical possibility that there are additional dimensions that remain small."

J. Polchinski, String Theory, volume I, CUP 1998 (p. 231)

My personal version of this argument is to say that in GR space can be curved and hence it is a "logical possibility" that there are dimensions so curved that they actually close on themselves. I find that it is a very seducing way to introduce the notion of extra dimensions that are hidden because of their compactness, since it follows from the well-established curvature of space-time.

Polchinski then goes on to explain Kaluza-Klein theory with a periodic dimension without further explanations. This leaves the following question open :

What kind of massive object curves some dimensions so as to make them compact while leaving other dimensions non-compact ?

I've been drawing a lot since then and realized a few things that amazed me for five seconds before revealing their complete triviality... Nevertheless, here are some of my reflections.

At first the idea that an object curves only some dimensions and not all of them seems bizarre. Imagine a toy universe that has only two dimensions so that one can envision curvature as the bending of an elastic membrane with a ball on it. This video which is part of a documentary about Hawking has a nice illustration of this after 3:00.

Now if some dimensions are to be so strongly curved that they become compact, this must be the effect of a very massive object. So let us give the ball a tremendous mass; the way I picture what happens is that the weight of the ball stretches the membrane in the form of an extremely long tube. So neglecting the extremities, the membrane has now the topology of a cylinder, which is the direct product of a non-compact manifold, a real line, with a compact one, a circle. What seems a bit weird is that the radial direction, which is curved by the ball, remains non-compact, whereas the angular direction, which is not curved, is compact ! But there is a fallacy here : to have an angular coordinate (with a finite range) does not mean there is a compact dimension. The plane is not any compacter in polar coordinates than in Cartesian coordinates...

I'd like to understand spaces that have a non-compact part and a compact part. For simplicity I'll take direct products of real lines and spheres. The simplest case is R x S^1. This is pretty clearly a cylinder if you think of it as an S^1 fibration over R, but you could also think of it as a R fibration over S^1, in which case one can see that is is also a cylinder by taking into account that the fibers must not intersect. In fact, only the topology matters here and the cylinder can be continuously deformed at will. The formal definition of this direct product is R x S^1 = {(x, theta) | x \in (-\infty, \infty), theta \in [0, 2\pi]}. So you just need to specify a couple of coordinates, one for the real line (x), and one for the circle (theta).

An interesting deformation of the cylinder is achieved through a conformal mapping, where one defines a coordinate z = exp(x + i theta). This can be thought of as a radial coordinate r = exp(x), and an angular coordinate theta. The radial coordinate ranges from 0 to infinity but at r=0 the circle degenerates, which does not correspond to the cylinder bounded by circles at both its extremities at infinity. In consequence, the origin has to be removed. So one ends up with the manifold R^2 (in fact the complex plane...) with the origin removed : R x S^1 ~ R^2\{0}. R^2 is non-compact but by removing a point you make it isomorphic to a space that is the direct product of a compact space with a non-compact space. There is thus an equivalence between manifolds with a singularity and partially compact manifolds !

Coming back to our latex membrane, we understand now that in order to obtain a compact dimension the ball must be so heavy that it produces a black hole singularity.

Now consider R x S^2. It is hard to understand as a fibration : is it some kind of solid cylinder ? I guess that a particle can move arbitrarily inside the cylinder but as it approaches the surface it is forced to move tangentially to it (?) It is much easier to think about it after a conformal mapping, which leads to R^3\{0}. Again, the origin is removed for the angular coordinates to be well-defined everywhere. (Notice that the manifold is not complex anymore, given that it is odd-dimensional...)

Another simple case is R^2 x S^1, which can be reduced to the first case by rewriting it as R x R x S^1 ~ R x R^2\{0}. We get a manifold R^3 with a line singularity along the third axis. This is nice because it means we can get more than just one non-compact dimension (the radial direction), namely the dimensions of the singularity itself.

We are ready to generalize : after conformal mapping, the direct product R^n x S^d gives a (n-1)-dimensional singularity at the origin of a (d+1)-dimensional space.

String theory requiring 10 dimensions for its consistency while we only observe 4 non-compact dimensions, we might try the direct product R^4 x S^6. However, according to our reasoning, this is isomorphic to R^3 x R^7\{0}, so that the fourth non-compact dimension differs from the others in that its boundaries at infinity are six-spheres... (how bad is that ?)

If this has to be avoided, one must rather consider R^5 x S^5 ~ R^4 x R^6\{0}. This is pretty close to what appears in the context of the AdS/CFT correspondence ! There, the singularity would be black 3-branes and there would be a warping between R^4 and the radial coordinate to get AdS_5.

Remark :

The bending latex representation is misleading since it implies that the membrane is curved towards a third dimension. I don't think this is the proper way to view it, because otherwise the curvature of four-dimensional space-time would by itself imply the existence of extra dimensions... Rather, one should maybe picture a curved space by a density plot. The darker the color, the larger the curvature. The surface of the black hole would be a very dark circle.

From the density plot the surface of the black hole is at a fixed radius, but from the curved membrane representation it forms a very long tube. It is folly to suppose the coordinate along the tube is the coordinate U = r/alpha' with r->0 defined by Maldacena.