Birkhoff's theorem should more precisely be formulated like this: if the Einstein field equations with a lagrangian L_M have a solution for a given spherically symmetric system, then the metric is static outside the system.
Birkhoff's theorem can be derived from the Einstein field equations in one page. One assumes a spherically symmetric metric which depends on the radius r and the time t. One can then show that the metric has to be independent of the time t in the vacuum zone surrounding the system.
Let us think of this. If we would have a lagrangian L_M which does not conserve energy, but creates more mass-energy into the system with increasing time, then the total mass of the system would increase with time and the metric outside it should change with time.
This means that if we assume general relativity & a lagrangian which does not conserve energy, then there is no solution for the Einstein equations.
General relativity seems to "know" that all lagrangians L_M must conserve energy. This is suspicious. Other force fields in physics do not "know" such things.
If the perpetuum mobile, which we sketched in our previous blog post, works, then general relativity itself does not conserve energy together with any lagrangian which allows the building of pressurized vessels. This would imply that general relativity is inconsistent with all realistic lagrangians L_M.
We do not know if classical electromagnetism conserves energy, because the self-force of a charge on itself is not well understood. This implies that it is not known if any realistic physical lagrangian is consistent with general relativity.
The proofs of the conservation of the ADM mass and the Komar mass probably assume that building a pressurized vessel is not possible. The proofs are probably correct if we assume just pointlike particles with Coulomb static electric and gravitational forces between them. Then the lagrangian probably does not allow the building of a vessel.
What is the problem with general relativity? If we think of a rubber sheet model, there we can affect the metric in the surrounding vacuum through operations which we do inside the system in the center. We can use pressure to increase the tension of the rubber in the surroundings of the system. No perpetuum mobiles are possible in a rubber sheet model. A consistent theory of gravitation probably must not imply Birkhoff's theorem.
What would a consistent theory of gravity look like?
Suppose that we would have a rubber sheet model where the weight of a mass can make a pit into the rubber and stretch it, but even an infinite pressure would be unable to maintain that pit if we remove the mass. Then the pressure can do as much of work as we wish, and we trivially have a perpetuum mobile in the model. Clearly, a consistent theory of gravity must allow pressure to stretch the spatial metric of spacetime.
The Einstein-Hilbert action calculates the local spacetime deformation energy from the Ricci scalar curvature at the spacetime point. How does that differ from a lagrangian for a rubber sheet model? In rubber, there is tension which is determined by the environment. Is it necessary that a consistent theory of gravitation has to include such a tension field? The concept of tension in spacetime sounds like an aether theory, which might be bad.
In the pressurized vessel thought experiment, it is the tension of the rubber sheet which resists the increase of the spatial volume of the vessel. It is hard or impossible to calculate the energy without a tension field.
Another question is if the tension has to extend past the wall of the vessel, to the environment outside. Probably yes. A steel ring (= the wall of the vessel) is not attached to the rubber sheet but rather it slides on slippery rubber without friction.
If we just have a weight resting on the rubber, the tension which it causes extends to the whole rubber sheet.
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