The contradiction from the Landau-Lifshitz pseudotensor
Kostas Kokkotas (2020) explains how an approximate metric for a gravitational wave packet is constructed. We start from a wave solution
g = η + h
of the linearized Einstein wave equations. We assume that g is a perturbation of the flat metric η.
Let us substitute g to the full Einstein field equations. It is not a solution, and in empty space we obtain a residual tensor. The residual tensor is red in the formula:
The red "stress-energy tensor" is the Landau-Lifshitz pseudotensor. It is supposed to describe the energy and momentum density of the wave:
Then we solve another perturbed metric
g' = η + h'
from the equation above. The full solution is then
η + h - h'.
That is, we form the metric far away by pretending that the wave packet is an electromagnetic wave of the energy E.
Let us have the space filled with initially static infinitesimal test masses m. When the packet W arrives, it pulls on the masses m. The test masses start to get denser around the path of W. That implies that R₀₀ is positive inside the wave W.
We arrived at a contradiction.
Could it be that the test masses in the path of W begin to "overlap", so that the density of any infinitesimal cube of test masses does not grow?
An overlap means that infinitesimal cubes of test masses will go inside each other. An overlap always develops when a long enough time passes. Paths of test masses from different sides of W will eventually cross each other.
It is hard to see how a weak gravitational wave could distort the configuration of the test masses so drastically, so that an "overlap" would develop immediately.
What is the problem in the Landau-Lifshitz trick? The problem is that it is sleight of hand which does not really solve anything. One cannot make the faraway metric to imitate that of an electromagnetic wave of the energy E without introducing a non-zero R₀₀ inside the wave.
The contradiction from Birkhoff's theorem
Let us have a spherically symmetric mass M. We assume that inside M we have devices which send and absorb gravitational waves. The devices cannot be exactly spherically symmetric, but we conjecture that Birkhoff's theorem still approximately holds.
The energy in the gravitational waves inside M cannot alter the metric far away, according to Birkhoff's theorem.
The conclusion is that the faraway metric generated by an almost symmetric configuration of gravitational wave packets W is just like for the analogous electromagnetic waves.
This suggests that an individual wave packet W has a faraway metric like the analogous electromagnetic wave packet. This leads to a contradiction like above.
The contradiction from the ADM formalism
Certain sources on the Internet claim that the ADM energy, which is defined as a surface integral of the gravity field very far away, is conserved. If the system is spherically symmetric, then this is Birkhoff's theorem.
Let us have a system M which sends a gravitational wave W and absorbs it back. If the claims about the conservation of the ADM energy are correct, then W must have a far-away metric which mimics an equivalent electromagnetic wave. We obtain a contradiction as above.
Exact solutions of gravitational waves
There are known exact solutions for gravitational waves such that R = 0 everywhere. But the waves then are plane waves, and are not physically realistic.
It looks like that realistic gravitational waves always break the Einstein field equations.
In our own Minkowski & newtonian gravity model, gravitational waves do not have a restriction R = 0.
Conclusions
It looks like that the Einstein equations do not have a solution for a gravitational wave packet.
Since we have empirically observed gravitational waves, this is yet another piece of evidence which suggests that the Einstein equations are incorrect.
Test comment June 17, 2024.
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