>I disagree that the "signatures" of the extended dimensions "requires" that
>there be one temporal and three spatial.
I think you may have misunderstood my meaning when I used the term
"signature". The signature *defines* how many dimensions are spatial and
how many are temporal. It is a technical term in the theory of
differential geometry. I explain further below.
> My reading of the hyperspace model
>(admittedly from the popular scientific press -- _Scientific American_,
>_Science News_, _Discover_ and the like -- including books like _Hyperspace_
>and _Cosmic Questions_) indicates that the theorists for the most part view
>the different dimensions as being equivalent. In fact many of them have
>crudely referred to these dimensions as "spatial", in the sense that the
>10-dimensional universe can be crudely visualized as a 10-dimensional
>hyper-object (hence the term "hyperspace") within 10- or 11-dimensional
>space. I have seen no attempt by any theorist to characterize these
>dimensions on the basis of any kind of "signature" or "flavor" or whatever
>else you want to call it.
As I understand it the leading superstring theories have a 10-dimensional
spacetime with a signature that requires that there be one temporal
dimension and 9 spatial dimensions. All six of the compactified
dimensions are (I think) *spatial* in character.
> As a result I could be wrong, but my impression
>is nonetheless that the dimensions of hyperspace in fact have all the same
>"signature" (or very nearly so) and that our characterization of "temporal"
>vs. "spatial" is subjective and naive based on perception and tradition, not
>hard science.
No. Time *is* physically different than space. Our perceptions cannot
detect differences between things for which there are no physical means
of distinction between them. IOW, if two things are physically
indistinguishable then our senses cannot perceive a difference between
them either. Spacetime has a Lorentzian signature (i.e. there is *1*
dimension of time and the rest of the dimensions are spatial). If it had
a Riemannian (or Euclidean) signature then all of the dimensions would be
spatial.
In the mathematical theory of differentiable manifolds which are endowed
with a metric (or sometimes called a *pseudo*metric to distinguish the
metric concept here from that of *metric spaces* where that notion of
metric includes a strict positivity requirement) there are two important
integers which are properties of the manifold. The first is the
dimension of the manifold. The dimension is effectively the minimum
number of real-valued coordinates that are necessary to label the
distinct points of (a sufficiently small topologically open and connected
subset of) the manifold. The second property is the signature of the
manifold which is the difference between the number of positive
eigenvalues and the number of negative eigenvalues of the metric tensor
at each point of the manifold. For the spacetime manifold of General
Relativity the dimension is 4 and the signature is 2. This is because
3 + 1 = 4 and 3 - 1 = 2. Here the 3 is the number of the *spatial*
dimensions and the 1 is the number of the *temporal* dimensions. The
eigenvalue of the temporal dimension is of the opposite algebraic sign as
those of the spatial dimensions, and this difference is what gives time
its uniquely different characteristics than those of the dimensions of
space.
The Lorentz signature of spacetime is a mathematical pain in that many
calculations and derivations are much more cumbersome (and less pretty)
than they would be if the signature was Euclidean (dimension=signature=4)
with 4 spatial dimensions and no temporal ones. Because of this often
workers in the field of theoretical high energy physics, general
relativity and cosmology often multiply the temporal coordinate by
i(=sqrt(-1)) which formally has the effect of changing the sign of the
temporal eigenvalue. This makes the theory formally look like it is for
a manifold of only spatial dimensions and the calculations become much
more tractable. In order to compare the results of the calculations with
actual experimental results in our universe they substitute back the
real time parameter in for the formal imaginary time parameter used to
simplify and symmetry-fy the derivations. It should be noted that this
trick needs to be used with caution because the topologies of the
symmetry groups for a Euclidean-signatured manifold are different than
those topologies for a corresponding Lorentz-signatured manifold, and
these topological differences can lead to errors for the unwary. In
fact, an example of this trick is behind Stephen Hawking's No-Boundary
proposal for the initial conditions of the Big Bang. When a Big
Bang-type spacetime is analytically continued over to imaginary time
then the inital BB singularity disappears in the Euclidean (4 space no
time) theory and this has certain suggestive implications for the
original spacetime theory.
>So if this is true, I ask again: how does the fact that the universe is
>made up of four dimensions affect Glenn's argument?
I'll leave this for Glenn.
>I should point out that I tend to agree with Dave on this issue. If in fact
>the universe is a 10-dimensional "hyper-object" it may have no more choice
>in what values to give to the universal constants than a cube has in
>determining how many faces it should have or at what angle those faces
>should connect.
>
>Kevin L. O'Brien
It should be remembered that at this stage it is still a big *IF* and a
*MAY*.
David Bowman
dbowman@georgetowncollege.edu