Janko,
Thank you for the questions. First for your questions: Yes, the r with the vector symbol is a unit vector. See page 186 of General Relativity:An Introduction for Physicists for the normal equation with the cosmological constant (but as the authors note does not fit with our understanding of current physics) also shows that there is no r3 (since the nature of an Omega term does not allow the volume to be modeled as a point).
I have not found a definitive derivation of the 8pi, although I have looked.
I had not run across the Hadley thesis and I find it an excellent read. I would have to go more in depth but his thoughts are similar to mine. What I would add in with his equation is that the expected value of the quantum vacuum is a "potential" energy that might be designated as a multiple of the metric but that there is no known method to combine this with the expected value of Guv.
I will look at the Petkov more in depth also.
Let me try another approach to explain this as compared to GR. In some ways, the whole problem also seems to stem from a notation problem.
Let us take the tensors for Omega and L, looking at this in an abstract manner without consideration of the meaning and derivation of Tuv yet, and let us assign some scalar values as a teaching tool. Looking at equation (3), assign the number 10100 to Omega, and let's refer to this as the curvature potential of spacetime. Then to G we will assign the curvature number due to matter viewed as a particle present at a point in spacetime, and to L we assign the curvature magnitude due to matter viewed as a wave at a point in spacetime. (apologies if a addition sign is missing) You can see that these would be of an opposite nature (can use either/or but not both at same time). [math]\Omega g_{\mu\nu}=G_{\mu\nu} L_{\mu\nu}[/math] From (4) we can see as G increases from zero, which is flat spacetime, the particle density must be increasing since the curvature [math]G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}[/math] is increasing. Likewise for the matter as waves, for flat spacetime Luv is of an equal but opposite value of Omega guv. When the wave density increases, Luv decreases in magnitude away from 10100 so that the curvature increases. [math]\Omega g_{\mu\nu} -L_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}[/math] You can view matter as a particle or a wave, but not both at the same time. This is what I mean by a notation mistake. The most general equation from the Riemann and metric is only written in the literature as [math]\Lambda g_{\mu\nu} G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}[/math] but this is bad notational form, since strictly speaking the tensor "Guv" in this equation does not technically have to equate to the Einstein tensor. It only is equivalent if the multiple of the metric in the equation is zero. Riemannian geometry should certainly have no preference for either way.
This explanation above might give someone who has studied deSitter space a fit. In deSitter space, there is no matter present so it is stated Guv is zero, but as explained in GR there is a multiple of the metric remaining which of course means that there is curvature, which is now linked to the cosmological constant problem and dark energy, which doesn't seem to make sense mathematically and hasn't yet been explained physically. In addition, MTW explains why simply adding the cosmological constant creates mathematical difficulties. In my view this is akin to epicycles, where new empirical evidence causes one to simply attempt to justify adding new features. While that may have worked for Neptune, I think it may be time to instead go back to the original equation and take another look.
This leaves several reasons for why, if I have a choice, to prefer the Luv with a multiple of the metric as waves.
1. The most general equation includes a constant of integration but there is no mathematical logic for preferring this to be zero, even with zero curvature. (Vacuum energy is a physical argument introduced by Zeldovich)
2. The Luv version allows this multiple of the metric to exist, even with a very flat spacetime as measured by WMAP.
3. If one estimeates what the vector field for the Newtonian spherical mass gradient would appear like through Luv, then it is of course an opposing gradient as the Newtonian one, but with the minus sign the gradients are mathematically equal. However, this should tend to make repulsive gravity seem like attraction at a distance.
4. The multiple Omega with the metric cannot be ignored in the Newtonian spherical mass gradient, but it would appear that this results in a point where the unit vector of the gravity switches to an opposite sign at a predictable point.
