On July 23, 2024 and August 2, 2024 we carelessly claimed that one can perform the Noether time variation by doing an infinitesimal metric variation in the space volume which is otherwise empty, except that it contains a gravitational wave.
We did not specify in detail what this infinitesimal variation of the metric is. Let us analyze it in this blog post.
The Noether time variation performed by stretching or contracting the metric of time g₀₀, either locally or in the entire spatial volume
We assume that we have a history H in the Einstein-Hilbert action, such that it converts the mass-energy of some matter to gravitational waves.
History H
###### /\/\/\/\/\/\/\/\/\/\/\/\/\/\/\/\ action = S
matter wave
----> t proper time
We do the Noether time variation by "speeding up time" by dt in the matter part of the history by dt and "slowing down time" in the wave part of the history. The variation is simply performed by adjusting the absolute value of the metric of time, |g₀₀|, somewhat larger in a segment of the wave history, and adjusting |g₀₀| a little bit smaller in a segment of the matter history.
History H'
#### /\/\/\/\/\/\/\/\/\/\/\/\__/\/\/\/\ action S'
matter wave
| | | |
time sped up time slowed
by dt by dt
----> t proper time
The Wikipedia variation for R in the Einstein-Hilbert actions says:
1. Increasing |g₀₀| in the wave part of the history H will not change the action integral there because R₀₀ = 0 (and R = 0). The Ricci scalar R remains zero in the variation.
2. Let the matter part be thin static matter of a density ρ. Then
R ≈ κ ρ,
where κ is the Einstein constant.
In the Einstein-Hilbert action,
R / (2 κ) + LM
≈ ρ / 2 - ρ
= -ρ / 2
is non-zero. The matter lagrangian density LM is of the typical form T - V, where T is the kinetic energy and V the "potential energy". That explains the -ρ above.
If we only change |g₀₀| locally, not in the entire spatial volume, then the Einstein-Hilbert action integral cannot change, since we assumed that it is a stationary point of the action for the matter.
But if we reduce |g₀₀| globally in the entire spatial volume of space, then the variation of R is zero. The volume element sqrt(-det(g)) becomes smaller. Thus, the value of the Einstein-Hilbert action integral changes in the matter part.
Is the Noether variation a variation of the metric or coordinates? Is it local or global in the spatial volume?
Can we do a Noether variation by varying the coordinates of events in a history?
We have to analyze what each type of a variation does in the above configuration.
We can speed up events either by varying the metric of time g₀₀, or varying the location of events in time.
Moving events in the matter part of the history to a slightly earlier time
Above we tried to vary just g₀₀, and realized that we would need a global variation to move all events in time. Maybe we can just move the events and make the variation local in that way?
In the matter part of the history H, can we vary the system in such a way that it starts to create the gravitational waves a coordinate time dt earlier?
The system might be almost static at an early stage of the history. The main feature of the metric might be an almost "settled down" Schwarzschild metric.
Let us vary the history H in such a way that the events at the almost static phase at the start happen a little faster. There might be no events at all, if the system is perfectly static.
1. The system starts creating gravitational waves a coordinate time dt earlier. The action integral probably loses a very simple contribution of a time segment dt in the almost static part of the history.
2. The initial phase of the wave part of the history H is moved back dt in the coordinate time. The action integral should be almost unchanged. The variation is local because the speed of light is finite.
3. The final phase of the wave part of the history H has the time progress of the waves slowed down, in such a way that the end state of H is not changed at all. Since the speed of light is finite, we only have to change the history H locally. This is equivalent to increasing |g₀₀| locally while keeping the coordinates of events fixed. Our reasoning in a preceding section above gives that the action integral remains zero if we modify g₀₀.
If we did not overlook anything, above we constructed a history H' through an infinitesimal local variation of H, such that its action integral S' differs from the action integral S of H. The history H is not a stationary point of the action.
If this is correct, the Einstein-Hilbert action does not allow creation of gravitational waves at all.
Discussion
The rogue variation which we invented in May 2024 shows that no dynamic system is a stationary point of the Einstein-Hilbert action. That is, general relativity does not have solutions for dynamic systems. This implies that there is no solution for any system which would produce a gravitational wave.
Our variation of the history H above is another way to show that general relativity cannot handle gravitational waves. This time, the problem is that it "hides" the energy of a gravitational wave.
Conclusions
It looks like general relativity cannot handle a Noether variation of the history.
We will next look at variations of the linearized (approximate) Einstein-Hilbert action. Maybe we will finally find the reason why the mass-energy density of a sine gravitational wave is the average of -R₀₀ / κ over a large spatial volume.
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