Kyro wrote:gmalivuk wrote:No, they're not.Kyro wrote:Someone please tell the string theorists that dimensions are orthogonal by definition.
If we're talking about something like curved manifolds (which we probably are when talking about space-time), you only need local homeomorphism to Euclidean space when talking about the dimension of that manifold.
And coordinate bases need not be orthogonal even when describing completely flat Euclidean space. Any set of linearly independent vectors can give a coordinate system for the space they span, which has a dimension equal to the number of vectors.
I should probably shut up before I realize I don't know what I'm talking about...
Even if you use non perpendicular vectors as the basis of your coordinate system, you still draw parallel grid lines. Therefore the dot product of the axes is 0. Therefore the dimensions are orthogonal.
Well, that simply isn't right. If you have non-perpendicular basis vectors, by definition their dot product is non-zero. So you wouldn't draw parallel grid lines. Doesn't stop them from being linearly independent, of course
It seems to me the notion of manifolds would suggest very large other dimensions, not small ones. I don't think the process is reversible.
The problem is this all vanishes when curvature comes into the picture, hence the qualifier "local." Think of the surface of a sphere: you can draw latitude and longitude vectors that are, at any given point, orthogonal to each other, but due to the curvature of the surface you have no unambiguous way of comparing a longitude vector at one point to a latitude vector at another.
On the topic of small dimensions, the easiest way to visualize this (as I think someone brought up earlier in the thread) is to imagine you're a tiny (and two-dimensional) ant on the inside surface of a cylinder. The cylinder has two dimensions, which you can look at as one going around and one going backwards and forwards. If the cylinder is really, really thin, you'll barely even notice the "going around" dimension - as far as you're concerned, you're living in a one-dimensional surface where you can only go back and forth. The second dimension is "curled up," much like the compact dimensions in string theory and other higher-dimensional physics theories. As you can see, there's nothing funny going on, much less anything in the definition of a manifold which precludes such a situation.
In response to Kyro (who posted after I started writing this), I hope that helps you understand why very small dimensions aren't necessarily nonsense!