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 :)