On 10/26/2012 8:33 PM, meekerdb wrote:
On 10/26/2012 4:55 PM, Stephen P. King wrote:
From the quote below: "it is expected that the 10-dimensional
space-time of string theory is locally the product M4×X of a
4-dimensional Minkowski space M3,1 with a 6-dimensional space X."
This "local product" operation, represented by the 'x' is the act
of adding two manifolds, one of 4 dimensions and one of 6 dimensions
for a total of 10 dimensions, thus this yields a very different
structure from, for example, a 10d Euclidean manifold.
All of the local degrees of freedom are present at every point
but the compacted ones are such that any motion (a translational
transformation within M^3,1) shifts from one local 6d manifold to
another 6d manifold. The 6d compactified manifolds are Planck sized
6d tori 'glued' (using the math of fiber bundles
<http://mathworld.wolfram.com/FiberBundle.html>) to each and every
point in the M^3,1 space. It is not correct to think of the compacted
manifolds (actually they are tori) as "free floating" in a 3,1
dimensional (not 4d for technical reasons as the signature of time is
not the same as the signature of the spatial dimensions) manifold.
They are manifolds - just some more dimensions that happen to be
compact. It makes no more sense to talk about them as 'free-floating'
than to talk about altitude free floating on lat-long; it's another
'direction', not an object.
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