Q' > Q.
How would electromagnetism react to such a magic trick?
The hamiltonian of the Coulomb field
A rudimentary "lagrangian" or, rather, a hamiltonian, calculates the energy of the (arbitrary) electric field in the space, where the energy density is
1/2 ε₀ E²,
and the electric potential energy of the charge Q is
-Q V,
where V is the electric potential. Incidentally, -Q has to be negative to simulate the field of a positive charge. The electric field allows Q to fall into a lower potential, releasing energy. The price we have to pay is that the creation of the electric field E consumes energy.
A stationary point of the hamiltonian, presumably, is the standard Coulomb electric field of a charge Q.
Magically adding more charge Q'
Let us then use a magic trick to grow the charge Q to a larger charge Q'. The system no longer is in a stationary point of the hamiltonian.
The Coulomb field of Q' would be a stationary point, but it differs considerably from the Coulomb field of Q, and differs from it in the entire Minkowski space.
How does the system develop after this? Without the magic trick, a dynamic system would pass between "adjacent" stationary points which differ from each other very little. But there is no rule about how the system should find a new stationary point after it was put very far from a stationary point by the magic trick.
We could let the field of Q' expand radially from Q' to all directions. Suppose that it now extends to a radius R. Since fewer field lines exist at r > R than at r < R, the system is as if we would have a shell of negative charge
Q - Q'
expanding.
But there is no such negative charge. The system is locally not at a stationary point.
If we try to put the system locally into a stationary point for all r < R and r > R, then we end up having a discontinuity of the electric field E at R. It is like a singularity, and is not allowed.
Another option is to switch between the Coulomb fields of Q and Q' either instantaneously, or slowly but synchronously, in the entire Minkowski space. That would allow superluminal communication and is forbidden by special relativity.
What is the value of the hamiltonian at different stages? If we simply add the extra charge to Q without changing the electric field E, then the value of the hamiltonian jumps suddenly. If we somehow are able to get Q' to descend down to a low potential, then the value of the hamiltonian decreases. However, this does not make sense in hamiltonian mechanics: the value should stay constant.
Birkhoff's theorem for Coulomb fields
If one assumes that the system only passes between adjacent stationary points, then the field outside Q cannot change. This is similar to Birkhoff's theorem.
If we then use a magic trick to add more charge into Q, there simply is no well defined path that the system should take. We could claim that the system ends up in a singularity, but that is just one option.
The analogy in pressure in gravity
What happens if we have a spherical mass M and we add some pressure into it?
The pressurizing operation, apparently, causes the system to move into a state which is very far from a stationary point.
The "Einstein-Hilbert action mechanics" may break down after that. There is no reasonable way to define how the system develops in time after the pressure changed.
One could try to solve the paradox by claiming that the pressurizing operation somehow requires a lot of energy since it takes the system far away from a stationary point. If the required energy is so large that one cannot change the pressure at all, that breaks newtonian mechanics.
The Ehlers et al. (2005) result
Ehlers et al. compensate an extra positive pressure inside a spherical vessel M with a negative pressure in the surface of the vessel. In electromagnetism, this corresponds to increasing Q to Q', and compensating the change in total charge by adding a shell of negative charge around Q'.
The change in the metric is isolated into M, and the problem which we described above does not occur. There is no need to update the metric in the entire Minkowski space.
The problem with pressure and gravity is known as Tolman's paradox. Ehlers et al. refer to R. C. Tolman's book (1934). In the book, in section 109 on Disordered radiation, Tolman writes:
"we are led to the interesting conclusion that disordered radiation in the interior of a fluid sphere contributes roughly speaking twice as much to the gravitational field of the sphere as the same amount of energy in the form of matter."
Conclusions
A problem in general relativity seems to be that it allows new "charge" to be created: pressure acts as a charge.
It is like the magic trick of creating new electric charge in electromagnetism. After the charge has been created, the hamiltonian or lagrangian does not tell us how the system should develop forward in time.
The conflict between Birkhoff's theorem and a pressure change would be a fundamental one: general relativity cannot describe the behavior of a system where pressure changes. Since there are pressure changes in all realistic physical systems, general relativity fails in every realistic case.
Our own Minkowski-newtonian gravity model does not treat the gravity caused by a pressure as a "field" or a "metric". But we have to check if the problem of a changing force field somehow causes other difficulties. For example, a perpetuum mobile might become possible if we can change the attraction of M. A rubber sheet model probably saves us from a perpetuum mobile.
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