Q-reeus said:
Yes, I see how those posts could be confusing. That's yet another thread that touched on this topic, and I'll have to go back through it to see how that whole discussion developed. That thread was quite a bit before the others. See further comments below on some specific items of confusion.
Q-reeus said:
As I say that was a shock to me. It has different physical consequences to what I now see your position is, which is that radial length coefficient only changes in shell transition, not transverse.
As long as you define "radial length coefficient" properly. As I've said before, it's important to pay careful attention to the distinction between coordinate-dependent quantities and actual physical observables. The "J" and "K" factors as I defined them in a previous thread are actual physical observables: "J" is the "gravitational redshift" factor, which can be observed by exchanging light signals with someone far away from the gravitating body, and "K" is the "non-Euclideanness" of space, which can be observed by counting how many small identical objects can be packed between two 2-spheres of area A and A + dA, relative to the Euclidean prediction.
It just so happens that in Schwarzschild coordinates, J = g_tt and K = g_rr; but that is a special property of that particular coordinate system, that its metric coefficients just happen to match up so nicely with physical observables. Since calling K the "radial length coefficient" depends on the fact that K = g_rr, that terminology is only really appropriate if you are using Schwarzschild coordinates; i.e., it is coordinate-dependent. Whereas the physical definition of K that I gave above is *not* coordinate-dependent; it may be harder to *express* K in terms of coordinate differentials and metric coefficients in other coordinates, but the physical meaning of K is still the same, and can be described without referring to any coordinates at all, as I did above.
Q-reeus said:
Thus for flat interior metric we must have that interior length scale is identical to that at infinity.
What do you mean by "interior length scale"? This looks to me like a coordinate-dependent quantity. The physical observable, K, being 1 in the interior vacuum region just means that, in that region, exactly as many small identical objects can be packed between two 2-spheres of areas A and A + dA as Euclidean geometry predicts. If that's all you mean by "interior length scale", then yes, it's the same in the interior as at infinity. But if you mean something else, you'll have to specify what you mean. The term "length scale" does not have a unique, well-defined meaning in GR, at least not if you are trying to use it to compared lengths in different parts of spacetime; it depends on the coordinates.
Q-reeus said:
Which means, because interior clock rate is rescaled as per J factor (or sqrt thereof), interior observer sees shell size as smaller than coordinate observer would.
You are correct that J < 1 in the interior vacuum region, so if one could look at light emitted from within that region that was somehow transmitted through the shell and received far away, that light would appear redshifted. That's the physical meaning of J.
Equating this to a different "clock rate" in the interior is OK, but relating that in turn to "shell size" depends on how you do the relating.
Q-reeus said:
Whereas your position back then amounts to interior observer seeing shell as larger to some extent - depending on just how the spatial components were meant to 'share' the change in transition. Anyway that provided the impetus for the later thread we have been recently referring to. All for naught.
In other words, "the size of the shell to an interior observer, compared to its size at infinity" is coordinate-dependent; with one choice of interpretation of coordinates, the interior observer sees the shell as "larger", but with another choice, it is seen as "smaller". None of this has anything to do with physical observables; the physical observable is that K = 1 in the interior vacuum region, and J < 1.
Perhaps I should adopt a policy of refusing to talk at all about coordinate-dependent quantities with you, since it seems to confuse you so much, and just stick exclusively to physical observables. But you in turn would have to agree to accept "sorry, that's coordinate-dependent, so there isn't a well-defined answer" as the answer to a *lot* of the questions you have been asking. You haven't shown much of a desire to do that.
Q-reeus said:
And in that later thread, as per previous quotes, I was led to believe there is this one-to-one match up between metric coefficients and SET components. Thus to explain g_tt shell transition change, T_00 only is involved, and for g_rr, T_rr only is involved, and so on. It is now evident that also is not so; K = g_rr has zero dependence on T_rr and purely depends on T_00, while J = g_tt has mixed dependence on T_00, T_rr, T_ss. From all this you may appreciate why I feel somewhat bamboozled - one might say somewhat misled.
Sorry if you were confused. Now that I've shown you the math, perhaps it will be clearer. But you have always maintained before that you were allergic to math.

Now that I know it actually makes sense to you, I'll feel less of an urge to try to translate into ordinary English, which as I've said many times, is fraught with inaccuracy.
Q-reeus said:
So anyway it now seems obvious stress in static shell does not have the metric altering properties I came to understand it was suppposed to have.
The metric, strictly speaking, is also coordinate-dependent; as I said above, it just happens to be a special property of Schwarzschild coordinates in spherically symmetric spacetimes that the "J" and "K" physical observables happen to match up exactly with g_tt and g_rr. In most cases that doesn't happen. It's much better to focus on the actual physical observables whenever possible.
Q-reeus said:
Which sort of makes any further discussion of that scenario pointless re this thread. Unless that is there is some sort of 'clean' (not mixing with other T terms) 1:1 match up involving radial and transverse stresses to curvature/metric terms that can be said to have well defined physical meaning?
I'll take a look at the curvature tensor components, which will show how physical observables associated with tidal gravity are affected. But again, in general, no, I would not expect there to be a "clean" match up between SET components and metric coefficients; first, since both, strictly speaking, are coordinate-dependent; but second, even looking at physical observables I would not expect there to be a "clean" match up in general.
Q-reeus said:
I was using the Wikipedia notation, which is a tau for total differential line element interval. Strictly one should use either ∂tau/∂r or ds/dr I suppose to avoid any confusion.
Wikipedia's notation is somewhat inconsistent; I've seen both dtau and ds used, and not always with attention to the metric sign convention either.
Q-reeus said:
So when all is said and done, this is just what I had thought back then, that dr/ds = K-1/2.
*If* you are using Schwarzschild coordinates. But that relationship is coordinate dependent; it is *not* true if you use a different radial coordinate. The physical meaning of the K factor is what I defined above.
Q-reeus said:
Well I maintain then the shell transition behavour of K is a purely mathematical consequence of adopting SC geometry and without sensible physical justification
No, the shell transition behavior of K, as I defined it *physically*, is directly observable; you can measure it wherever you like, in principle, by testing how many small identical objects can be packed between 2-spheres of area A and A + dA, and comparing the result to the Euclidean value. This can be done, in principle, inside the shell, and the result will change as you go down through the shell from outer to inner surface. There's nothing "mathematical" about it. It's a physical observable.
The only "mathematical consequence" here is how the physical observable, K, shows up in the metric, since that depends on the coordinates. But that doesn't change the actual physics; it only changes how the physics is represented in the math. It seems to me that you are creating a lot of confusion for yourself by failing to grasp this basic point.
Q-reeus said:
since in my view, J = 1/K should hold in all regions involving shell. But that's my view and gets us away from this thread topic.
Well, it would be interesting if you could give some actual physical reason why J = 1/K (with J and K defined as physical observables, as I did above) should hold everywhere, instead of just in the exterior vacuum region.