Dear John Merryman:
I'm sorry that I got sidetracked from these comments over the past couple of weeks. In two posts above, on Sep. 17, 2012 @ 10:42 and on Sep. 17, 2012 @ 10:49, you wrote:
`What is the source of that "rate of propagation through space as a constant value," if it isn't space? The most basic measure of cosmic distances is how far light travels in a year. So you say "space expands." Therefore it's not that expanding space. What determines how you move that marker down that elastic band, that defines constant? If it is "constant," then it must mean it's stable. What is that stable frame????'
and
`"Galaxies stay the same size and move apart from one another in an expanding universe."
`What about the space essentially collapsing into these galaxies? The reason Einstein came up with a cosmological constant in the first place was to balance gravity and keep the universe from collapsing into a point, due to gravity, so why does cosmology now treat galaxies as inert points of measure in an expanding universe?'
First of all, the space around galaxies actually isn't supposed to be collapsing into them; the situation isn't so much like Einstein's initial reason for proposing the cosmological constant as I think you're getting at. The problem that Einstein was having was that in a finite, but unbounded universe (think of a two-dimensional universe, as a two-dimensional surface of a sphere), the gravitational attraction amongst a uniform distribution of galaxies would cause space itself to contract as all the galaxies would come closer together. In such a universe, even if it was expanding, that gravitational attraction would eventually cause the expansion to halt and re-collapse. On the other hand, if (isotropic and homogeneous) space were infinite and flat, or negatively curved, the mutual gravity of all galaxies wouldn't be enough to halt expansion, so it would continue forever.
But I don't think this is really what you were looking for. The problem, I think, is that you're thinking of space in the near vicinity of a massive body, as collapsing into that body. That isn't the case. It's supposed to be warped towards such a body, like a trampoline with a bowling ball on it, so that test-particles (with much less mass, so that they don't have any appreciable effect on the curvature of space in comparison) fall into it. The trampoline itself isn't collapsing into the bowling ball in this case; it just has that shape for all time. (Technically, it's space-time that's supposed to be warped about the massive body according to general relativity, but this idea captures the picture well enough.) Now, according to standard cosmology, this is roughly the discription in the vicinity of any galaxy or bound cluster of galaxies, where the gravitational attraction is supposed to dominate over the expansion of space that's taking place everywhere there's sufficiently empty space. So it's as if the bowling balls hold space fixed out to a certain point, beyond which the fabric of the trampoline is stretching. The reason why it's appropriate to simply use the FLRW models which approximate bound masses as points, is that these regions where space isn't expanding (or contracting, but remain warped and fixed) are so minute compared to the expansion that's taking place on cosmological scales.
Now, what about in the early Universe, when galaxies would have been far more densely distributed? The matter in the Universe was far more uniformly distributed to begin with, as evidenced by the CMBR, and expansion was everywhere (and has remained, except in small regions where enough matter is concentrated) an after-effect of the big bang. As space expanded, gravitational seeds, where there was slightly more mass, attracted local matter against the expansion of space, and mass concentrations formed, near to which space would not be expanding, though in between which space would continue to expand, etc.
Therefore, Eddington's description of dispersing desks is appropriate: the galaxies remain the same size because the local gravity dominates over the expansion of space that takes place between galaxies.
I think that should answer your second question. The answer to your first question is that it isn't just space through which the invariant speed of light is defined, but space and duration, and the metrical structure of space-time, which is the graduating map of events that occur in enduring space. The RW metric is a space-time metric with null lines that describe the rate of propagation of light through space in time. The null lines are invariant, so it doesn't matter in which frame you measure the speed of light, the magnitude is always the same.
I think the reason you're having trouble with this is that you're thinking of the speed of light through space, with time passing in a Newtonian sense that maybe doesn't so much enter into the idea explicitly. Then, because space is expanding, you're wondering how light can travel through it always at the same speed. The answer is that according to relativity theory, light travels along null lines of the space-time metric. It's this metric that provides the stable measure that you're asking about.
An interesting case to consider is de Sitter space. It's a hyperboloid of one sheet in Minkowski space, so that a two-dimensional slice of de Sitter space can be depicted in three dimensions. A hyperboloid of one sheet is a doubly-ruled surface: there are two straight lines passing through every single point. These lines are actually the null lines of de Sitter space. (The surface t=-infty in Fig. 1 in my essay is actually a pair of such null lines, extending in opposite directions along the hyperboloid.) Now consider the cosmological model described by Eq. (3) in my essay: the three-dimensional enduring universe is the three-dimensional sphere that contracts from T=-infty to T=0, and then expands thereafter. In this space-time, regardless of whether space is expanding or contracting, these straight null lines are consistently well-defined at every point in space-time. Therefore, at any value of cosmic time, T, the spatial component of the velocity of light is also well-defined. As you should be able to see clearly by looking at that figure, the coordinate velocity of these null lines through the three-sphere is much greater when it's small, so that while null lines subtend an angle pi from T=-infty to T=infty, the majority of that is actually covered near T=0.
I hope these answers are helpful. Sorry again that it took so long to respond.
Best, Daryl
