On 10/25/2012 11:52 AM, meekerdb wrote:
On 10/25/2012 4:58 AM, Richard Ruquist wrote:
Stephan,
Since yesterday it occurred to me that you may be thinking of the 10
or more dimensions of string theory as being orthogonal because they
were so before the big bang. But the dimensions that
curled-up/compactified went out of orthogonality during the big bang
according to Cumrun Vafa. I'll look up that reference if you are
interested.
According to Vafa 2 dimensions compactified for every single space
dimension that inflated. In over simplified terms, 2 dimensions
(actually in strips of some 10,000 Planck lengths) to be compactified
lined up say in the east-west space dimension so that space in an
orthogonal direction could expand. So some semblance of orthogonality
exists in the compactification process, but it is clear that the
compactified dimensions become embedded in 3D space for inflation to
occur.
It's implicit in the definition of dimensions of a Riemannian manifold
that there are as many orthogonal directions as dimensions.
Compactified dimensions are just small; they're small, not infinite,
because they have closed topology. That property is completely
independent of having orthogonal directions.
Brent
Dear Brent,
Compactness and orthogonality are not the same quantities. Yes. But
my point is that the compact structures in string theories (super or
not) are orthogonal to the dimensions of space-time. Maybe we need all
take a remedial math class on linear algebra and geometry!
--
Onward!
Stephen
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