The forum discussion centers on the validity of the shell theorem in General Relativity (GR) and its relationship with the Schwarzschild solution. It is established that while the second part of the shell theorem holds in GR, the first part is not meaningful due to the absence of a consistent GR solution for the spacetime geometry surrounding a point mass. The discussion emphasizes that the Schwarzschild solution is derived under the assumption of spherical symmetry and a vacuum solution, and does not rely on treating a spherically symmetric object as a point mass.
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Actually it has to be greater than 9/8 of the Schwarzschild radius, per Buchdahl's Theorem.PeroK said:The Schwarzschild solution is valid outside an object, as long as the radius of the object is greater than its Schwarzschild radius.
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.Bosko said:Do you think that for ##r_1 < r_s## the mathematical model does not correctly represent physical reality?
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:PeroK said:For ##r < r_s##, the coordinate ##r## is timelike.
PeroK said: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.
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.Bosko said: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.
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.Bosko said:There is no parameter of a spherically symmetric object in the solution. For example, the radius .