UPDATE August 3, 2024: The argument below is flawed. See the note which we wrote to the beginning of the July 23, 2024 blog post. The variation of the metric g has to have a continuous time derivative. To satisfy that, we may have to extend the variation back inside the mass M. The Ricci tensor R is not zero there. We have to analyze what happens in the variation there.
See our blog post August 4, 2024. Our claim below is correct, but the proof a little different.
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Thus, the variation of the action S is not zero for an emission process of a gravitational wave. The energy is visible before the emission, but invisible in the wave. Energy conservation is broken.
In the Einstein-Hilbert action, an allowed physical history is a stationary point of the action S. For a stationary point, any local infinitesimal variation must leave S unchanged. But an infinitesimal time variation by dt changes the value of S for an emission of a gravitational wave.
We proved that no emission is an allowed history for the Einstein-Hilbert action. That is, general relativity does not allow the existence of gravitational waves.
Any transfer of energy through gravity breaks the Einstein-Hilbert action?
In our reasoning above, the crucial point was that R = 0 in space which is empty of matter (gravity fields are allowed in that space).
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M ● • m
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v
Suppose that we move a mass M back and forth, and through the gravity force of M, accelerate another mass m far away. We transfer kinetic energy from M to m.
The kinetic energy cannot be transferred infinitely fast. Some of it must be on the way from M to m.
But, according to the Einstein field equations, the Ricci tensor R = 0 in the space between M and m. A Noether variation of time is not able to discern the energy in the gravity field. Thus, energy is not conserved, and the history is not allowed by the Einstein-Hilbert action.
How to fix general relativity?
The obvious fix is to treat gravity as an ordinary force field, not as a special field which "alters the geometry of spacetime". Our own Minkowski & newtonian model should have no problem in handling energy transfers through gravity.
Butcher, Hobson, and Lasemby (2009) start from linearized gravity and add a self-coupling to the mass-energy of the field. We have to check how they handle the energy transfer problem.
Conclusions
We found yet another major flaw in general relativity. It is well known that defining energy is problematic in general relativity, but, apparently, no one before us has noticed that it spoils the Einstein-Hilbert action.
The main point of lagrangians is to prove conservation of energy. Actually, we should have expected that the Einstein-Hilbert action cannot work.
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