# Re: Compact dimensions and orthogonality

Dear Richard,

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. i.e. space-time.

On 10/26/2012 6:36 PM, Richard Ruquist wrote:
The requested excerpt from
http://www.scholarpedia.org/article/Calabi-Yau_manifold#Calabi-Yau_manifolds_in_string_theory:

"Calabi-Yau manifolds in string theory
Superstring theory is a unified theory for all the forces of nature
including quantum gravity. In superstring theory, the fundamental
building block is an extended object, namely a string, whose
vibrations would give rise to the particles encountered in nature. The
constraints for the consistency of such a theory are extremely
stringent. They require in particular that the theory takes place in a
10-dimensional space-time. To make contact with our 4-dimensional
world, 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 . The 6-dimensional space X would be
tiny, which would explain why it has not been detected so far at the
existing experimental energy levels. Each choice of the internal space
X leads to a different effective theory on the 4-dimensional Minkowski
space M3,1 , which should be the theory describing our world."

The 6d space is tiny indeed, said by Yau in his book "The Shape of
Inner Space" to be 1000 Planck lengths in diameter. The rest of that
reference apparently describes a number of possible realizatons of the
6d space that is way beyond my comprehension. So now I am reading
http://universe-review.ca/R15-26-CalabiYau.htm, a math review of Yau's
book,
to get a more definitive answer to our questions.
Richard.

On Fri, Oct 26, 2012 at 4:48 PM, Stephen P. King <stephe...@charter.net> wrote:
On 10/26/2012 4:31 PM, Richard Ruquist wrote:
Yes

http://www.scholarpedia.org/article/Calabi-Yau_manifold#Calabi-Yau_manifolds_in_string_theory
Hi Richard,

Could you cut and paste the specific description that answers Brent's
question?

On Fri, Oct 26, 2012 at 3:01 PM, meekerdb <meeke...@verizon.net> wrote:
On 10/26/2012 5:08 AM, Richard Ruquist wrote:
No Roger,

In string theory dimensions are conserved but can undergo extreme
modification such as in compactification where formerly orthogonal
dimensions become embedded in 3D space in spite of what Brent thinks.

Do you have a reference that describes this 'embedding'?

Brent

--
Onward!

Stephen

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