Today, August 9, 2024, we added a note there which says that we can change the value of the "wave part" of the action if we simply vary g₀₀ infinitesimally there. This proves that no history H can produce gravitational waves with the linearized action. A total failure.
If we ban arbitrary variations of the metric g, but do allow an undulation of the metric in a gravitational wave (especially, g₁₁ and g₂₂ are allowed to undulate from 1 slightly), then we might be able to have legal histories H which create gravitational waves. The restrictions would bring the linearized model closer to a conventional field theory, which has canonical coordinates, and does not play with metrics.
But still we would have that the energy density of a sine wave varies locally between negative and positive. It would be very strange if a perturbation of flat space can somehow carry a negative energy. It would lead to all kinds of paradoxes.
As we have remarked several times in earlier blog posts, the fact that a sine gravitational wave shortens certain distances, may allow superluminal communication, and is not acceptable, since it brings the plethora of time paradoxes. On August 5, 2024 we wrote that to get rid of the superluminal problem it may be necessary to treat the metric as a side-effect of an underlying "mass polarization" field. The idea of specifying the physical system through a metric may be hopeless.
A gravitational wave with zero energy can turn into positive energy when absorbed by matter?
Let us have the metric g of a positive spherical very small and thin mass M with a shell of a negative mass ≈ -M. We adjust the masses in such a way that the outside metric is flat.
There is obviously energy contained in the metric g, because g is not flat, and g can interact with matter, making the matter to move or vibrate.
What does a Noether time variation say about the energy content of the metric g?
Inside M, let the mass-energy density be ρ. The Ricci tensor is approximately like the following, if we set κ = 1:
R =
ρ / 2 0 0 0
0 ρ / 2 0 0
0 0 ρ / 2 0
0 0 0 ρ / 2.
The Ricci scalar is R = ρ and R₀₀ = ρ / 2.
If we "slow down time" by dt by making |g₀₀| larger, then g₀₀ declines from -1, and g⁰⁰ = 1 / g₀₀ increases from -1. The value of R increases. Our reasoning on August 5, 2024 says that the "energy" of the metric is then negative there.
Let us consider the metric of the M & -M system and assume that the mass densities are set zero. The "energy" of the metric g is approximately zero.
The metric cannot be static, as we removed the mass densities. The metric, presumably, starts to undulate. If we put test masses there, they can gain positive energy from the undulation of the metric.
Negative "energy" turns into positive energy? That does not sound sensible.
Pressurized vessel whose spatial metric is initially stretched
In this blog we have tried to construct perpetuum mobiles by assuming an extremely rigid vessel filled with incompressible fluid, such that the interior volume of the vessel is initially stretched with a mass M and the Schwarzschild metric around M.
If we suddenly remove M, an immense pressure is created. The calculation of Ehlers et al. (2005) suggests that the pressure is infinite. That is not sensible. The energy of the perturbation of the spatial metric would be infinite.
The Ricci tensor element R₀₀ is very close to zero if we only manipulate the metric of space, not the metric of time. If the "energy" of a metric perturbation is determined based on R₀₀, the value cannot really reflect the immense energy hiding in the stretched spatial metric.
Discussion about negative local energy
Defining the "energy" of a metric through the Ricci scalar R looks hopeless at this stage.
In a reasonable physical model, perturbing the metric from flat should always carry a positive energy?
If we perturb a rubber sheet from its equilibrium position, does the perturbation always carry a positive energy?
push push
---> <---
---------------------------------- tense rubber sheet
"negative
energy"
Not necessarily. If the sheet is tense and stretched, we can pull it with our fingers, so that it at some patch becomes less stretched and contains less elastic energy.
The "negative energy" of the shrunk patch is compensated by the positive energy of the stretched parts of the sheet.
One might be able to define "negative" energy as the energy missing from the default elastic energy of the stretched rubber sheet. However, this may not help in the case of gravity, since there are so many problens in defining a reasonable energy for a metric perturbation in general relativity.
Conclusions
Defining a reasonable energy density for a metric perturbation in general relativity is hard, if not impossible.
In our own Minkowski & newtonian gravity model, the "metric" is just a side-effect of the underlying gravity field. The side-effect can interact with matter, and does influence the energy density of a gravitational wave. But it may be that defining the energy density of a perturbation cannot be done in s beautiful way from the metric. We have to define it based on the underlying gravity field.
The "invisible polarization of masses" model of our August 5, 2024 post (section Conclusions) may help us to build a model for waves in complex interactions which have various side effects.
An example: if we have a crystal, then sound waves are simple displacement waves of atoms, but the interactions associated with these displacements may be very complex. In the case of a crystal, the "invisible polarization" is, actually, real atoms. It is kot invisible.
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