Re: Topology of the universe

In article <[EMAIL PROTECTED]>, Ralph Hartley
<[EMAIL PROTECTED]> writes: 

> An interesting article:
> 
> http://www.nytimes.com/2003/10/09/science/09COSM.htm

Just a comment, since it might not be obvious to all readers, and since 
as usual this will probably be painted by much of the media as a 
"sensation" which "shakes the foundations of physics" and possibly even 
"challenges Einstein".  The first I'll agree with---if the hunches turn 
out to be true.  The second and third, certainly not.

The range of "standard cosmological models", i.e. those based on General 
Relativity and or those generally referred to as Friedmann-Lemaitre 
models, have something called the curvature parameter.  This, however, 
refers to the LOCAL curvature, not the global topology, about which such 
models say nothing.  (In other words, with classical cosmological tests 
(i.e. working out the dependence of an observable quantity on redshift 
and fitting the cosmological parameters to the observations), one can't 
determine the global topology---one needs other methods, such as 
analysing the CMB.)

The "local" curvature can be the same everywhere---and has to be if the
model satisfies the cosmological principle.  An analogy is a cylinder:
its local curvature is 0, i.e. the same as a plane.  Its global
topology, however, is obviously different.  One can see this via the 
fact that one can roll up a sheet of paper to make a cylinder without 
having to stretch, tear, fold or wrinkle it.  In more than two 
dimensions, things are more complicated, but the principle is the same: 
a given local curvature can exist in more than one global topology.