Demurral and Request for Clarification
yogi said:
what I detect was a shift in the direction of attempting to explain physics in terms of space as a substantive
You should say "I think I detect", since others familiar with his writings can and do disagree about what he appears to have meant by various statements.
yogi said:
in the spherical space-time universe of de Sitter, the two sphere surface grid is composed of time and space.
Huh? As you know there are many coordinate charts we can use on various pieces of the de Sitter lambdavacuum, aka H^{1,3} (a Lorentzian spacetime with uniform negative curvature), including comoving charts adapted to static (non-inertial) observers and to various families of inertial observers. In particular,
<br />
ds^2 = \frac{-dt^2 + dx^2+dy^2+dz^2}{(t/a)^2}, \;<br />
0 < t < \infty, -\infty < x, \, y, \, z < \infty<br />
is conformal to the upper half space portion of Minkowski spacetime and is comoving with a "collapsing" family of inertial observers whose world lines form a vorticity-free timelike geodesic congruence, whose orthogonal hyperslices are all isometric to euclidean three-space. Here, intervals of t do not correspond to intervals of proper time as measured by ideal clocks carried by our observers, but we can change to a comoving chart
<br />
ds^2 = -d\tau^2 + \exp(-2 \, \tau/a) \; \left( dx^2 + dy^2 + dz^2 \right),<br />
\; -\infty < \tau, \, x, \, y, \, z < \infty, \;<br />
There is a very similar chart for a family of
expanding inertial observers, who also have orthogonal hyperslices isometric to euclidean three-space. But there are also families of inertial observers whose world lines form a vorticity free timelike geodesic congruence, whose orthogonal hyperslices are each isometric to H^3, e.g.
<br />
ds^2 = -dT^2 + a^2 \, \sinh(T/a)^2 \; \left( \frac{dx^2+dy^2+dz^2}{z^2} \right), \;<br />
0 < T, \, z < \infty, \; -\infty < x, \, y < \infty<br />
And there are observers who have constant acceleration directed radially inward in a suitable polar spherical chart, such as this one (analgous to Eddington chart):
<br />
ds^2 = -(1-(r/a)^2) \, dv^2 + 2 dv \, dr + r^2 \; \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right), \;<br />
-\infty < v < \infty, \; 0 < r < \infty, \; 0 < \theta < \pi, \; -\pi < \phi < \pi<br />
The world lines of these observers again form a vorticity-free timelike congruence whose orthogonal hyperslices are each isometric to S^3. Many other charts can be found in Hawking and Ellis and in the literature, e.g. "static" version of the previous chart (analgous to Schwarzschild chart), Brill chart, Penrose chart, etc. I stress again that there are both expanding and contracting observers in this spacetime who can be used to define "negligible density" cosmological models (better say "toy models").
So what is your "two-sphere grid"?
yogi said:
As the spherical universe expands, both the time dimension and the space dimension increase in relation to their previous lengths ...its the inflating balloon model except that the surface is composed of one time dimension and one space dimension rather than two space dimensions - so the reference for change is the previous surface itself
This makes no sense as written. Can you explain what you are trying to say in terms of the line element in some coordinate chart? Any "increase" or other physical change will probably be seen to refer to some family of observers whose motion can be characterized geometrically, independently of coordinate description.
yogi said:
If you like - with intense effort I will probably to be able to understand about every third word
That's not very encouraging!
Yogi, may I ask: what is your comprehension rate when you read Hawking and Ellis? If you don't understand timelike congruences and their kinematic decomposition (acceleration, expansion scalar, shear tensor, vorticity vector) and null geodesic congruences and their optical scalars, you really can't hold any meaningful discussion of cosmological models! Remember, this ideas were imported/introduced more than forty years ago and immediately became standard core topics due to their great utility. These techniques are perfectly suited to studying the geometry (i.e. the physics) without getting confused by "features" ("bugs"?) which merely characterize a particular aspect of a given coordinate representation.