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If we start with minkowski spacetime in 4 dimensions and then add several curled up spatial dimensions attached at every spacetime point, then:

I'll label a spacetime point as:

(ct,x,y,z)[a1,a2,a3,..,an]

where the bracketted coordinates are the 'curled' coordinates.

- If we label the coordinates of the curled dimensions as [0,0,..,0] where they attach to the minkowski spacetime point, does that mean you can't move from point (ct,x,y,z)[anything] to (ct',x',y',z')[something] without going through (ct,x,y,z)[0,0,..,0] and (ct',x',y',z')[0,0,..,0] ? Or am I misunderstanding what is meant by "attach at every spacetime point"?

- While spacetime is still continuous instead of discrete, it seems like there is still a preferred frame now: The frame in which the 'density' of the curled up dimensions in each direction is equal. After doing a boost, the 'density' of the curled up dimensions is greater in one direction. This, in my mind, is analogous to the famous example of the twin's paradox in a closed universe ... while there is of course still no local preferred frame, there is a global preferred frame now. Since strings are "global" in the sense that they can go all the way around a curled dimension, wouldn't they make such a global preferred frame acutely apparent?

What I'm trying to think out comes down to something like this (poorly worded, I know, but hopefully gets the point across):

How can local poincare symmetry in the full dimension spacetime still yield "apparent local" poincare symmetry after "coarse graining" over the global structure of the curled dimensions?

Can anyone help me understand better? (even just correcting my terminology here, or giving me terminology to use, to make the discussion more precise would be very helpful)

Thank you.

I'll label a spacetime point as:

(ct,x,y,z)[a1,a2,a3,..,an]

where the bracketted coordinates are the 'curled' coordinates.

- If we label the coordinates of the curled dimensions as [0,0,..,0] where they attach to the minkowski spacetime point, does that mean you can't move from point (ct,x,y,z)[anything] to (ct',x',y',z')[something] without going through (ct,x,y,z)[0,0,..,0] and (ct',x',y',z')[0,0,..,0] ? Or am I misunderstanding what is meant by "attach at every spacetime point"?

- While spacetime is still continuous instead of discrete, it seems like there is still a preferred frame now: The frame in which the 'density' of the curled up dimensions in each direction is equal. After doing a boost, the 'density' of the curled up dimensions is greater in one direction. This, in my mind, is analogous to the famous example of the twin's paradox in a closed universe ... while there is of course still no local preferred frame, there is a global preferred frame now. Since strings are "global" in the sense that they can go all the way around a curled dimension, wouldn't they make such a global preferred frame acutely apparent?

What I'm trying to think out comes down to something like this (poorly worded, I know, but hopefully gets the point across):

How can local poincare symmetry in the full dimension spacetime still yield "apparent local" poincare symmetry after "coarse graining" over the global structure of the curled dimensions?

Can anyone help me understand better? (even just correcting my terminology here, or giving me terminology to use, to make the discussion more precise would be very helpful)

Thank you.

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