TGlad said:
if there is no matter present there can be no particles.
Test particles don't count as "matter"; they are particles that are so small that their stress-energy is negligible, so they don't contribute to spacetime curvature. This term occurs all the time in GR textbooks and papers, so it's important to understand what it means and how it is used.
TGlad said:
if it is clearer then let's talk about some imaginary test particles that have zero mass (or as mass tends to zero if that is preferred).
That what I
was talking about. See above.
TGlad said:
Instead of a ball of them, we have a 4D block of particles with dimensions (defined as the distance from one side to the other) of δt,δx,δy,δz\delta t, \delta x, \delta y, \delta z.
No, you don't. You are still not grasping a key point:
coordinate intervals are not distances or times. That is a very important aspect of GR that many people find hard to grasp.
Once again, go back and read the Baez article you linked to, carefully. It doesn't mention coordinates once. Even ##t##, which might otherwise be mistaken for a coordinate, is carefully defined to be the
proper time of the particle at the center of the ball. Proper time is actual, measured clock time, not coordinate time.
Also, the point of Baez' article is to show you how to translate the Ricci tensor, or more specifically the component ##R_{00}## of the Ricci tensor, into something physical which is easily visualized. That requires considering, not just one infinitesimal 4-d volume of spacetime, but a whole series of them, describing the history of the ball of test particles--what happens to it over time (proper time of the particle at the center of the ball) because of the effects of ##R_{00}##. In other words, what Baez is describing, in 4-d spacetime terms, is a "world tube"--a 4-d volume that is like a cylinder or tube, with a small spatial "width" and running along the "time" direction (but here, again, "time" means proper time, not coordinate time).
TGlad said:
We could call x the proper distance or the coordinate distance along the axis that has block width ##\delta x##, it shouldn't matter at ##e## because that is the rest frame of the particles, so they coincide.
No, they don't.
Also, ##e## is an event, not a frame. They are not the same thing.
It looks to me like you lack some important background in GR. What textbooks, papers, or other resources have you studied.
TGlad said:
I will do more reading up to try and better understand the Ricci-flat (zero covariant divergence) constraint.
These are not the same thing. The condition of zero covariant divergence applies to the Einstein tensor, not the Ricci tensor, and it applies in all spacetimes, regardless of their geometry; it is a geometric identity called the Bianchi identity.
Ricci flat means the Ricci tensor is zero. Only certain spacetimes satisfy that property.