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Obviously the question's title has an unspecified subtext: intuitive to me.

Some background to pitch the discussion appropriately: I have a broad understanding, more qualitative than quantitative, of the Lagrangian formulation of the standard model. My General Relativity is pretty vague. I have a pop-science grasp of string theory, where this particular question comes from.

I understand that in order to explain certain aspects of string theory, a number of extra dimensions in addition to our observed 3+1 are required (possibly because there are local gauge invariances that only work with higher dimensions? I'm vague on the details). It's assumed that if these extra dimensions were broadly similar as the three spatial ones we experience, their presence would be very obvious as even classical laws would be drastically different to how they actually are. To reconcile this, it is said that they are compact.

The question concerns compactification, in several aspects:

  • Is there a specified mechanism by which this is thought to occur? If so, is this mechanism the same as, or related to, the mechanism of 'warped spacetime' from GR?
  • Can these compact dimensions be understood as variations of the spatial dimensions in all but their geometry, or are they fundamentally 'different'? That is, if they were to be de-compacted just a little bit, could we start to observe real particles moving in those dimensions?
  • Is the state of a compact dimension similar to that of the observed 3 (spatial) dimensions shortly after the Big Bang? (This seems to tie into #1 as my understanding is that the small size of the early universe was due to extremely warped spacetime)

Some mathematical rigour would be appreciated in any answers, but be gentle - please :)

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I warn that my understanding of string theory is roughly the same level as yours, i.e. pop science. But the notion of "compact" as I understand it really is pretty much that which you would get in a topology textbook: "compact dimensions" are sections of spacetime that are closed and bounded (to use the Heine-Borel style description of compact, which we can do since manifolds are an atlas of charts each locally like a copy of $\mathbb{R}^N$).

The $x,\,y,\,z$ spatial dimensions are often taken to be "noncompact" dimensions because a section of constant $y,\,z,\,t$ in a chart is like the real line - both ends going off to infinity.

In contrast, a compact dimension would be for which the relevant section is closed and bounded. This needfully means it is homeomorphic to a circle $\{e^{i\,\phi}|\,\phi\in\mathbb{R}\}$. You could run off along the $\phi$ axis and eventually you would get back to your beginning point. And the distance you would need to traverse to get back to your beginning point in the compactified dimensions of string theory are very small: so the idea is that we simply don't see them from our everyday scales.

Have a look at my other attempt to write something meaningful on these ideas here.

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.

"Flat" and "noncompact" are two notions that are related by the Bonnet-Myers Theorem. A Riemannian or Semi-Riemannian manifold of everywhere positive Ricci curvature is compact. In Lie theory, a related idea is that a Lie group with a negative definite Killing form (and a few other fiddly conditions) is compact. The Myers theorem gives you a bound for the "diameter" of the manifold in question - the "flatter" it is, the "bigger" it is - and the bound is saturated if and only if the manifold is isometric to a constant curvature sphere.

See an interesting discussion at MathOverflow on what the fundamental group of the Universe might be like

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My knowledge of string theory is cursory, but I do know the basics of Kaluza-Klein reduction, on which string theory is based. The compact dimension in Kaluza-Klein theory is spatial, and the manifold is $M\times S^1$, where $M$ is the 4-dimensional spacetime and $S^1$ the circle. The extra dimension is different from the other spatial ones in that it corresponds to a Killing vector and its radius is extremely small, otherwise is just like any other spatial dimension. The smallness of the radius is such that we cannot observe particles move in 4 dimensions. That the fifth dimension corresponds to a circle is just an assumption, it is not related in any way, as far as I know, to curvature due to energy.

The introduction of the fifth dimension is just an assumption Kaluza made almost a century ago. It has the nice side-effect that the Einstein-Hilbert action with the aforementioned assumptions in dimension 5 yields the Einstein-Hilbert action plus the Maxwell term in dimension 4, which many physicists think that is more than just a mathematical trick, enough that thousands of articles and careers have been based on this.

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Jerry Schirmer's answer to "Why are extra dimensions necessary?" seems (as far as I can tell) to give a good explanation of why compact dimensions are compact, from a QM point of view. I think it is more or less equivalent to the GR idea that mass/energy creates space and that the energy needed to create big space in higher dimensions is so much that it just doesn't happen.

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