Date: Apr 25, 2013 7:34 AM
Subject: Re: closed universe, flat space?

> Let?s look at spacetime.  According to GR, it is the curvature of
> spacetime that causes gravity.  So, spacetime is curved around a
> gravitating mass.  In free space, the Einstein tensor vanishes which
> means the Ricci tensor also vanishes which mean the Riemann tensor
> also vanishes.  So, you have vanished Riemann tensor in curved
> spacetime.  That means the curvature tensors really do not address the
> curvature thing.  The field equations are merely differential
> equations that allow you solve the local geometry and nothing more.
> <shrug>
The Ricci tensor vanishing does not imply the Riemann tensor does .

Also ,don't confuse space with space-time . Space is only a 'slice' of
space-time . In itself it can be flat, while still being in a curved
space-time . The metric of space is three-dimensional , and embedded
in the four dimensional metric of space-time. Even factoring out the
gravitational effects (which predominantly affect the time-time
component of the metric tensor, not it's space components ) ,we still
have to consider the expansion of the universe .

Space can be predominantly flat (provided it's free from strong
gravitational effects) , while still being embedded in a curved space-
time .

> Write down the metric for n-sphere please.  <shrug>

If the earth were perfectly round , you would return to your point of
origin when going around it . Generalize . Or better yet, use Google
to figure out the metric .

> At least, you admit your own version of cosmology is purely
> speculation.  The so-called experts believe in their speculated
> ?reality? more whole heartedly.

First, don't underestimate the value of rational speculation .I
speculate that the sun will rise tomorrow .
All science has origin in rational speculation.
Second, the most commonly accepted model of the universe is flat (in
space) and infinite.Observations confirm
the universe 'flat with only a 0.4% margin of error' .

I find it much easier to read your post with this image in mind

If you post had a point to make, I don't see it. <shrug>