UPDATE October 8, 2021: The book Gravitation of Misner, Thorne, and Wheeler solved this mystery. Section 35.12, formula (35.54) says that the ripples of gravitational waves do alter the background geometry like mass-energy does. Using the rubber membrane analogy of gravity, a wave stretches the membrane a little bit, just like a weight placed on the membrane.
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In our October 7, 2021 blog post we asked if gravitational waves break general relativity by giving mass-energy to ordinary matter and creating a new "source" of gravity.
It looks like that exactly is the case.
A photograph of Albert Einstein by Oren Jack Turner, 1947. https://en.wikipedia.org/wiki/File:Albert_Einstein_1947a.jpg
A basic principle of the Einstein field equations is that only matter can act as a "source" of the field. Matter contains mass-energy, momentum, shear stress, and pressure. All these appear as sources of the Einstein gravity field in the equations. The gravity field itself is not a source.
A gravitational wave does not contain a source. It is a wave of the homogeneous (vacuum) Einstein equations.
But the wave can interact with matter, make the matter to vibrate and to heat up. Thus, the wave can create a new source of gravity.
) ) ) ) ●
gravitational lump of matter M
wave packet
Assume that we have a lump of matter in an asymptotically Minkowski space. Its mass-energy is M. We let a gravitational wave packet to hit the matter and increase its mass-energy to M'.
Assumption 1. We assume that a "reasonable" solution for the Einstein equations exists for this configuration. That is, there is a solution where the mass M is surrounded by an approximate Schwarzschild geometry and the wave packet looks like wave packets in electromagnetics.
If there is no solution, then the Einstein equations are too strict - there probably are only solutions for artificial symmetric configurations, like the Schwarzschild solution.
Let us look at the development of the gravitational field after the wave hits the matter and increases its mass-energy.
Far away, the field
G M / r²
suddenly changes to a stronger field
G M' / r².
If we look at the lines of force of the field, it looks like this:
-------- --------
M' ● --------
-------- --------
stronger weaker
field M' field M
Assumption 2. We assume that there exists a solution where which describes the temporal behavior of the system, and where the mass M' is surrounded by an approximate Schwarzschild geometry, except that far away, the geometry is still for M.
Contradiction. Lines of force seem to end in empty space. There is a source (or rather, a sink) of the field in empty space which does not contain matter - a contradiction.
We know that gravitational waves exist and that they contain energy. Waves remove huge amounts of mass-energy from binary pulsars and merging black holes. The process is the reverse to what we described above: a source of gravity M suddenly becomes a lot smaller source M'.
If gravitational waves themselves would be a source of gravity, then we would have no contradiction. Then our thought experiments would only involve moving mass-energy to a different place.
There are no longitudinal waves in electrodynamics. If charge were not conserved, then we could create a longitudinal wave by letting a charge q vary in time. We believe that there are no longitudinal waves in gravity, either. Then we must have conservation of the source of gravity.
Is it possible to save Einstein's idea that gravity is described as the geometry of spacetime? Can geometry itself act as a source of geometry?
How general relativity might be saved (and is saved)
In our diagram, the gravitational wave does not contain any source of gravity, but it may still be the case that far away it bends spacetime in the same way as the mass-energy difference M' - M would do. Is this possible?
___
___/ \ __________ drum skin
\___/
wave ---->
----------__ __----------- drum skin
\ /
● weight
If we have a tight drum skin, waves in it do not bend the skin in the same way as it would bend if we put a weight on the drum skin.
Gravitational waves are derived from linearized Einstein equations. In the case of electromagnetics, the wave does not behave like an electric charge.
Weak gravity has little nonlinearity. Electromagnetic waves presumably are an excellent approximation of gravitational waves.
Looking at the literature, it turns out that gravitational waves do bend background geometry just like mass-energy does. Thus, general relativity is saved from the contradiction.
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