Quote by PeterDonis
I think you're confusing the metric and curvature in the exterior vacuum region with the metric and curvature in the nonvacuum "shell" region. Everything DaleSpam and I have been saying applies only to the metric and curvature in the nonvacuum region. In the exterior vacuum region the stressenergy tensor is zero; there is no pressure or energy density, so there is no "matter contribution" at all. The curvature of spacetime in the exterior region is determined by the overall mass of the shell, via the form of the Schwarzschild metric, but that metric is a solution of the *vacuum* Einstein Field Equation (stressenergy tensor equal to zero). The exterior metric does not depend on any details of the shell's internal structure; two shells with identical overall mass, but drastically different energy density (say one is much thicker than the other, and correspondingly much less dense), would lead to the same metric in the exterior vacuum region. But within the shell, the metric would be quite different for those two cases: the spatial metric would have to change to flat over a much shorter distance for the thin shell, so the curvature components would have to change must faster as you descended through the shell.

Here's where the issue seems to be hitting the fan. I entirely meant shell matter's contribution to the exterior, SM region, and said so explicitly in #25. So no confusion on my part there. Agree with essentially all the rest above, but not the probable implication that stress in the shell can account for anything remotely significant re transition through shell wall. There is a severe logical chasm imo. Exterior, *presumably* anisotropic SM generated by essentially shell matter exclusively. Transition to interior flat region demanding 'steep' gradients in at least one spatial metric component. Relative pressure arbitrarily small in selfgravitating case for 'small' shell. Are we trying to pull a rabbit out of a hat, folks?
Now this would all go away if in fact hte SM is *actually* isotropic spatially, so are SC's fooling us? This is why I want that test sphere result so bad! Bed time.