I Missing proof of the Shell theorem in General Relativity

[PeterDonis]

The Schwarzschild solution is valid outside an object, as long as the radius of the object is greater than its Schwarzschild radius.

Actually it has to be greater than 9/8 of the Schwarzschild radius, per Buchdahl's Theorem.

 

[PeterDonis]

Do you think that for $r_1 < r_s$ the mathematical model does not correctly represent physical reality?

That's not the issue. The issue is that you don't understand what physical reality the mathematical model represents for values of $r$ less than $r_s$. For such values of $r$ the model represents the interior of a black hole. It does not represent the vacuum region outside an ordinary gravitating object with that radius.

 

[PeterDonis]

For $r < r_s$, the coordinate $r$ is timelike.

This is true for Schwarzschild coordinates, but there are other charts for which it is not true. You give a much better description of the actual problem (which can be stated in a form that's invariant, so it doesn't actually depend on your choice of coordinates) here:

The region of spacetime represented by $0 < r < r_1 < r_s$ for any $\phi, \theta$ and fixed $t$ is not a spatial volume. It cannot be the volume of a massive object.

 

[PAllen]

There is no parameter of a spherically symmetric object in the solution. For example, the radius .
The solution does not depend on the size of the object.
Let's find some detailed explanation of the formula and see when the radius of the object vanishes .

Is this from Wikipedia good for analysis?
https://en.wikipedia.org/wiki/Derivation_of_the_Schwarzschild_solution

I'm looking for a detailed derivation available on the internet.

Can we agree that the solution does not depend on the radius of a spherically symmetric object?
I think that this is some form of the shell theorem.

You assume nothing but spherical symmetry, and vacuum outside some r. You assume nothing about inside r (except spherical symmetry throughout). You derive that the solution is unique up to one parameter - a mass parameter associated with everything inside r. Thus you have derived, not assumed, that nothing about the nature or size of what is inside r can matter. There is simply no place for it in the solution. The shell theorem in GR is thus derived, not assumed. It is also different from the Newtonian statement ( you can’t talk about a mass point). The GR statement is: if all vacuum outside of r, nothing about the inside matters except one parameter describing total mass.

 

[Orodruin]

There is no parameter of a spherically symmetric object in the solution. For example, the radius .

There is no object at all in the Schwarzschild spacetime. It is a vacuum solution. The assumptions are spherical symmetry of the solution and vacuum.