Why are 11 dimensions needed in M-Theory? The four I know (three spatial ones plus time) have an intuitive meaning in everyday life. How can I think of the other seven? What is their nature (spatial, temporal, ...) and is there an intuitive picture of what they are needed for?
Answer
This should be a comment since I am not a string theorist but its too big. When Luboš (Luboš correct me if I'm wrong) speaks of the "shape" in his comment:
They're spatial dimensions - new temporal dimensions always lead to at least some problems if not inconsistencies - but otherwise they're the same kind of dimensions as the known ones, just with a different shape. To a creature much smaller than their shapes' size, they're exactly the same as the dimensions we know. The theory implies that the total number of spacetime dimensions is 10 or 11. We don't have "intuition" for higher-dimensional shapes because the extra dimensions are much smaller than the known ones but otherwise they'er intuitively exactly the same as the known spatial dimensions
he means that the higher dimensions are "compactified". A simple example of a compact space is a circle, or a Cartesian product of circles (a torus) or a high dimensional sphere. The crucial idea is that they are topologically compact, meaning roughly that they are finite and closed i.e. they have no boundary just like the torus or sphere have no boundary. So Luboš's little creatures would return to their beginning point if they walked far enough in the same direction.
As I understand it, one of the proposed "shapes" for the compactified dimensions is the Calabi-Yau manifold. Wholly for gazing on beauty's sake, its worth also looking these here at the Wolfram demonstrations site. Be aware that you're looking at a projection, hence the seeming "edges" are not the manifold's boundary. Like the torus and the sphere, these manifolds would let a little creature return to their beginning point eventually by travelling in a constant direction and nowhere would they come across a barrier or boundary.
Actually, it's not out of the question that the three spatial dimensions of our wonted experience are like this too, just that we're talking awfully big distances (10s to 100s of billions of light years) for us to come back to our beginning points if we blasted off into space and kept going in the same direction. As I understand this, this idea is seeming less and less likely since our universe globally is observed to be very flat indeed. See an interesting discussion at MathOverflow on what the fundamental group of the Universe might be like
Update: See also this answer clarifying some of my description of compactified dimensions. If they are big enough (as for our everyday three spatial dimensions, if they are compactified too) even though a constant direction vector can be integrated to a closed loop through space, the fact that the Universe is expanding means that one cannot traverse this loop in a finite time.
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