Hi Ben,
Thanks so much for the questions! I will attempt to return the favor in your thread but I am afraid that my questions may not be as pertinent as I am unfamiliar with some of the terminology in your essay. Will do my best though.
As for your questions here;
1. It may be that understanding positive curvature doesn't necessarily mean attractive gravity is a bit counter intuitive. Perhaps I should have prefaced this statement. My assumption is that, simply due to symmetry, the full equation should be able to work equally well with Guv or Omega guv-Luv, they both would give the same positive curvature. Curvature increases positively as Guv increases from zero, or as Luv decreases from the value of Omega guv.
The fascinating thing is though, the approximation to a Newtonian field gradient does not produce the exact same field equation. A symmetry breaking of classical gauge theory if you will. Without a quantitative analysis, it is plausible that this difference would only be detectable at larger radii. It also appears that to approximate this effect with only Guv one would have to add in a very tiny multiple of the metric (dark energy).
So what the 5th point probably should state is that if Guv and Omega guv-Luv are both suitable for the EFE, but only Guv requires an extra multiple of the metric that also breaks the condition of zero curvature with no matter, then Omega guv-Luv should take precedence over Guv-Lambda guv.
What this also means is that positive curvature can be equated to either attractive gravity or repulsive. Guv and Luv should result in very different field magnitudes of Phi. The Newtonian gravitational force, however, only depends on the spatial gradient of this field. The gradient vectors of Phi from Guv and Luv are of opposing directions of course, but vectors from Guv and -Luv are equivalent. What is even more interesting is how a constant multiple of the metric becomes a very tiny gradient vector opposing this.
Here is a small thought experiment and I apologize for the lack of mathematical rigor but it should show you what I mean. Take two vectors A and B, of equal large magnitude but they are placed tip to tip opposing each other. Their vector sum is a point, or a vector of zero magnitude. Now subtract from this sum a vector C pointing left that is smaller than either A or B. The total equivalence of all three vectors is a vector D that is of equal magnitude of C but pointing to the right. Occams Razor states that the simplest answer is only D, but the physical reality may be closer to A, B and C. Parsimony also requires that we choose the simplest anti-derivative to produce Newtonian gravity, and that we choose the simplest tensor for the Einstein field equation. Similarly, as vector D increases, you are increasing an attractive force, as vector C increases, you are decreasing a repulsion.
If I can parallel transport D on a curved surface, can I not also do the same for the sum of A, B and C? My guess is probably not and that we have missed something in manifold theory, although I am a long way from deriving this.
2. (Sorry 1 was so long!) My argument is that gravity is causal. The simplest way to understand what I mean is to state how I think gravity works. I think that matter at a point reduces the available field energy in a sphere surrounding it. The symmetry of the matter also depends on the symmetry of the reduced field energy. (this is not energy in the regular definition of matter-energy, more of a curvature potential) If another point within this sphere of reduced field energy also contains matter, it too reduces the available energy within a certain radius. The matter of these two points interfere with the symmetry of each, and both seeking symmetry (related to principle of least action) is manifested in a force where to tend to move together. The equations for the Newtonian approximation, however, seem to show that if clumps of matter were to spread far enough apart (past the constant within the EFE) then it would result in a purely linear repulsion. This would seem to fit a theory where the accelerating expansion only occurs after enough average distance between clumps of matter but it seems counter to our understanding of "expansion of space".
3. For Dark Matter, suppose that Omega guv can vary within a region not centered on baryonic matter. I suspect that this would add a stress energy tensor to the field equation. Since this tensor did not result from the exact point that matter is located, then the only way we would be able to detect it is through gravitational effects of matter passing through this region. I am a long way from deriving a set of christoffel symbols to account for this though.
Regards,
Jeff
