The discussion centers around the validity of the shell theorem in the context of General Relativity (GR), particularly regarding its proof independent of the Schwarzschild solution. Participants explore the implications of the shell theorem in GR, contrasting it with classical Newtonian physics and addressing the derivation of the Schwarzschild solution.
Participants do not reach a consensus on the proof of the shell theorem in GR or the derivation of the Schwarzschild solution. Multiple competing views remain, particularly regarding the interpretation of the shell theorem and the assumptions underlying the Schwarzschild solution.
There are unresolved issues regarding the assumptions made in the derivation of the Schwarzschild solution and the implications of Buchdahl's Theorem on the stability of configurations in GR. The discussion highlights the complexity of transitioning from Newtonian to relativistic frameworks.
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 .