re: Topology of the Universe

On Sunday, October 12, 2003, at 02:57 PM, S-P & M-M Sirag wrote:

Jack,

You say,

    "In particular I do not as yet see any physical necessity for
Saul-Paul's Ansatz that physical 3D expanding accelerating cosmic space is
actually SU(2)/ID rather than the SU(2) shown in Fig 3..."

    The idea that 3D cosmological space SU(2)/ID rather than SU(2) is the
view of Luminet et al. in their *Nature* (9 October) paper. They say (p.
593-594):

    "The Poincare dodecahedral space is a dodecahedral block of space with
opposite faces abstractly glued together, so objects passing out of the
dodecahedron across any face return from the opposite face. Light travels
across the faces in the same way, so if we sit inside the dodecahedron and
look outward across a face, our line of sight re-enters the dodecahedron
from the opposite face. We have the ILLUSION [emphasis added] of looking
into an adjacent copy of the dodecahedron."

Yes, but I have a hard time understanding how to translate that clear 
formal statement into physical operational informal terms of what an 
Argonaut with Alcubierre warp drive /\zpf field super-technology would 
actually experience on a fast trip across space now when c/Ho ~ 10^28 cm?

Suppose Kirk & Crew shoot across a distance ~ 10^28 cm in say one week 
proper time. Since they are, at all points on their worldline, on a 
local free-float timelike geodesic of their own making, Alcubierre 
points out that there is no necessary huge time-dilation twin effect as 
in global special relativity. That is, they can return in ~ 1 week of 
Earth time. So now the Mission is to forge "straight ahead" not to do a 
180 turn around back to Earth. With your SU(2)/ID which Lou agrees with, 
they do so continuously and get back to Earth anyway! OK maybe it works 
that way. But maybe it does not? That is an empirical issue that cannot 
be described by pure mathematics alone. The issue here is how to tie the 
pure math to the actual experiences. Perhaps they do approach an "edge" 
or a "wall in space" that is a giant wormhole and they can see Earth 
straight ahead through the wormhole mouth. It's a big mouth of course of 
order c/Ho diameter. I suppose you would say, "Well the mouth is so big, 
that there is no real operational difference in the two descriptions." 
Perhaps that is correct.  How could we pin down the specific finite 
topology of SU(2)/ID of 120 spherical 3D snugly tileddodecahedrons for 
example. I mean suppose Kirk's ship is tracked by radar as an FTL object 
on what appears to us to be a spacelike worldline even though locally on 
the ship it is timelike geodesic. Will we see 120 images of the star 
ship at some point? Or something like that? We must be able to detect 
SU(2)/ID in some way for the Star Ship voyage for it to make physical 
sense and to tell it is not some other topology?

Luminet writes "so objects passing out of the dodecahedron across any 
face return from the opposite face" and I have yet to understand 
physically what exactly that formal idea means. What does it mean for 
"an object to pass out of any face the dodecahedron" and to then "return 
from the opposite face"? The two faces have a co-moving separation 
c/H(t) does the object "return" at the same t as in a Star Gate or not? 
 Also maybe we can have it return at t +- delta t for a time machine 
effect with some tweaking.  What does the word "face" mean physically, 
or operationally, in Luminet's writing? That is the essence of my 
question of interpretation of the math here I suppose.


    "Poincare dodecahedral space" as they describe it is exactly the 
same as
SU(2)/ID.  These authors are quite aware of this equivalence as they show in
the paper "Cosmic microwave background constraints on multi-connected
spherical spaces", which you can download from:

     http://xxx.lanl.gov/abs/astro-ph/0303580

    On page 2 (of this 5 page paper), they say:

    "The finite subgroups of S^3 are the cyclic groups Zn, the binary
dihedral groups D*m, the binary tetrahedral, octahedral and icosahedral
groups, respectively of order n, 4m, 24, 48 and 120."

    Now since you are looking for a physically significant difference
between SU(2)/ID and SU(2) tiled with 120 spherical dodecahedra, I should
point out the fundamental significance of the volume of the 3D space, which
affects both the light travel times as well as the density of the space.  As
these authors say:

    "In all cases the volume of the space S^3/G is the volume of the
3-sphere S^3 divided by the order |G| of the holonomy group."

    From my point of view there is much more physics to be derived from the
structures S^3/G, and that has to do with the A-D-E classification of these
spaces. This is because A-D-E Coxeter graphs classify at least 20 physically
relevant mathematical objects.

    As I have mentioned previously, these include Coxeter (reflection)
groups, Lie algebras,  gravitational instanton spaces, catastrophe bundles,
2D conformal field theories, Heisenberg algebras, and much more.

Agreed, how do we get physically testable ideas out of that math? That's 
a rhetorical question at this point. I don't expect you to have definite 
answers.


    I have started writing this up in a Word file, and will send it to you
as soon as I finish it.

    Saul-Paul

Cool.

I am still not clear on how to physically interpret Luminet's Fig 3.