On 10/25/2012 10:49 AM, Richard Ruquist wrote:
On Thu, Oct 25, 2012 at 1:43 PM, Stephen P. King<stephe...@charter.net> wrote:
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!
I am still waiting for the explanation of how you know that to be true-
that the compact manifolds are orthogonal to space dimensions.
Richard
If they weren't orthogonal then a vector on them could be represented by by a linear
combinations of vectors in 3-space - and then they wouldn't provide the additional degrees
of freedom to describe particles and fields. They'd just be part of 3-space.
Brent
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