In the link, Donald Salisbury (2019) tells us about these early stages.
Chris Hirata's derivation of the energy density
Chris Hirata (?) (2019) presents a simple way to derive the energy density carried by a gravitational wave. First construct a wave packet solution
g = η + h
for the linearized Einstein equations. There, h is a small perturbation of the flat Minkowski metric η.
If the metric g = η + h were a solution of the full, nonlinear, Einstein equations in the vacuum, then in
we would have the stress-energy tensor T on the right zero. But η + h is not a solution. We obtain a non-zero value for T. The component -T₀₀ supposedly tells us the energy density of the gravitational wave described by h.
Does this make sense? Can we modify g slightly, in such a way that g' would be an approximate solution for the nonlinear Einstein equations? If yes, then why would the improved solution g' carry the energy in -T₀₀?
Let us make a metric perturbation k such that
η + k
is a solution for the Einstein equation for the above stress-energy tensor T. Then
g' = η + h - k
is an approximate solution for the full nonlinear Einstein equations in the vacuum (if such a solution exists). We assumed here that the Einstein equations are approximately linear.
We may assume that -k contains the "background metric" associated with the gravitational wave packet. That background metric corresponds to the stress-energy tensor -T.
Let us then assume that the large-scale metric of an isolated system S embedded in asymptotically Minkowski space cannot change. That is, the gravitating mass-energy is conserved.
If a subsystem of S creates a wave packet h and loses the energy E from its stress-energy tensor, then the metric -k far away must be equal to the metric created far away by E. Otherwise, the large-scale metric would change. It makes a lot of sense to treat -T₀₀ as the energy density of the wave packet h.
Does the energy density prevent faster-than-light communication inside a gravitational wave?
In this blog we have claimed that a gravitational wave h could break the speed of light limit c in the underlying Minkowski space. That is because if h temporarily shortens the distance between observers A and B (or speeds up the local metric of time), then A and B could exchange information surprisingly quickly.
Now we understand that the energy contained in the wave h slows down the local speed of light, because of the large-scale metric in -k above.
Is -k able to slow down light enough, so that we cannot get superluminal communication?
Apparently not. In the Hirata lecture (2019), the energy density of h is proportional to the square of the time derivative of h. If we make the amplitude of h very small, then its energy content is negligible, and the large-scale metric does not slow the local speed of light enough.
Did we just prove that gravitational waves clash with special relativity? If yes, then the Einstein equations describe gravitational waves incorrectly!
No. We forgot to take into account the gravity field of the mass M which is the source of the gravitational wave. The mass M slows down the local speed of light by
~ 1 / r,
where r is the distance from M. If h would increase the local speed of light by ~ 1 / r, the speed can still stay below the global Minkowski speed of light.
Let us move a small mass M back and worth. Its gravity field lags behind: this is the origin of a gravitational wave. The stretching of the radial Schwarzschild metric around M is
1 + 1/2 rₛ / r,
where r is the distance from M and rₛ is the Schwarzschild radius. The slowdown of time in the Schwarzschild metric is
1 - 1/2 rₛ / r.
The lagging gravity field creates a wave which "transports" a part of this metric perturbation away, as well as distorts it.
The amplitude of the wave goes as ~ 1 / r. The wave at some large distance r might shorten a local tangential distance by a factor
1 - 1/4 rₛ / r,
but since the local time runs slower by a factor
1 - 1/2 rₛ / r,
the local speed of light is still slower than in the surrounding Minkowski space. Faster-light-communication in Minkowski space is not enabled.
Our reasoning indicates that gravitational waves without a source mass might break the light speed limit c in Minkowski space. But are such waves possible?
Let us imagine a localized wave packet which starts from a certain position X far away from any masses. The packet travels a long distance, spreads, and gets absorbed by various masses M. If we play time backwards, we have those masses M producing a tight wave packet at a distant location X. Is the gravity of the masses M strong enough at the faraway location, so that it prevents faster-than-light signals?
Generating a gravitational wave with rods and pressure
We can harvest energy very efficiently from a gravitational wave if we utilize a rigid rod between two masses M. The wave alters the distance of the masses, and we can extract a lot of energy from the pressure inside the rod. Just let segments of the rod slide past each other with a lot of friction. Energy from the gravitational wave is turned into heat.
This rod system might be lightweight enough, so that we are able to break the speed of light at the location X of the previous section?
Most probably yes. The gravitational wave passes past the rod very fast. We do not even need any masses at the ends of the rod, because the rod does not have much time to move. If the rod is almost infinitely rigid, it can harvest a considerable amount of energy from the gravitational wave, while being very lightweight. Such a material does not exist in nature, but in principle it could exist.
Let us assume that we have a lightweight rod system which is able to harvest almost all energy out of a gravitational wave packet. If the play the process backwards in time, our rod system creates the wave packet.
As the wave packet passes two observers A and B, the distance between A and B, the packet probably makes the local speed of light faster than c in the background Minkowski metric. This open ports for a time machine, and the paradoxes which arise from it.
We probably have here a contradiction in general relativity. What is the origin of the contradiction? It is in the assumption that we can describe gravity with a metric. We have earlier in this blog claimed that a metric most probably cannot capture all features of an interaction.
In our own Minkowski & newtonian gravity model, a photon can only gain more inertia in an interaction. An interaction can only make the photon to move slower. Thus, our own model does not suffer from the problems described above.
Conclusions
Earlier in this blog we have argued that a gravitational wave in general relativity can speed up light. That leads to a time machine and paradoxes in special relativity. We cannot accept such behavior.
In this blog post we analyzed the process in more detail. To create a rogue gravitational wave, one must use pressure in a very rigid rod. Moving masses probably cannot do the job.
We have here more evidence that the concept of a metric is flawed. It cannot describe gravity.
But general relativity does predict correctly that the energy density of a gravitational wave is 16X the analogous electromagnetic wave. We have to find out why.
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