If it involves perturbing coordinates I immediately smell coordinate artifacts. Also, without any 4-curvature surely they must be non-propagating i.e not wave solutions always having gradients.
Feel free to name them "coordinate artefacts", this is close to what they are considered to be in GR spacetime ideology, where their existence in reality is denied. Once we cannot measure them, they do not exist.
As I said, try it out, learn how to compute the curvature, then compute the flat metric $\eta_{mn}$ in different curved coordinates, compute the curvature tensor in these different coordinates, and find out that it is really zero, even if the $g_{mn}(x,t)$ look very nontrivial, with quite arbitrary gradients, in such curved coordinates.
Look like non-propagating? LOL. Try it out, with coordinates $x^m \to x^m + f^m(x,t)$ with four functions $f^m(x,t)$ which look as much as propagating waves as you like. Simply use solutions of the standard wave equation $\square f^m(x,t)=0$ for them. Let's try:
\[ \eta_{mn} dx^m dx^n \to \eta_{mn} d(x^m+ f^m(x,t)) d(x^n+ f^n(x,t)) \approx \eta_{mn} (dx^m dx^n + \partial_k f^n dx^k dx^m + \partial_k f^m dx^k dx^n + \ldots) = (\eta_{mn} + h_{mn}(x,t)) dx^m dx^n .\]
Now, if the $f^m(x,t)$ are harmonic waves, the $\partial_k f^m(x,t)$ as well as sums of them are such harmonic waves too. So, the $h_{mn}(x,t)$ too (approximately, which, as you have acknowledged, is not a problem).
And these are the four types of gravitational waves which in non-covariant theories are nontrivial gravitational fields, but in GR are valid but trivial fields. With zero curvature tensor (in all theories).
That Living Reviews article plainly states that GR admits only TT waves (far field obviously).
Oh, something new, you use some mainstream GR review as the ultimate authority, and refuse to check it, and to learn to understand why this is claimed. Once you accept it as an authority, why not also about the claim that in GR such waves exist?
And, just to clarify: The other waves cannot be nor created nor detected, but are
admitted in valid solutions of GR as well.
The assumption being such are generated by real matter sources having fluctuating multipole moments - beginning with quadrupole.
There is a general principle applicable to any type physical wave - EM, acoustic, GW, thermal, whatever. Any source must also be able to function as a detector - at least in principle. Your ghost waves (if actually waves not static/quasi-static entities) do not satisfy that criteria thus irrelevant to OP.
Fine. This is what I have tried to explain you. In GR, they cannot be generated by anything, and cannot be detected by anything. That's why they can be ignored as irrelevant. And, with sufficient positivism, which rejects the very existence of unobservables, even as nonexistent. Or one can start name-calling them "coordinate artefacts", but even this does not change the formulas which describe them.
This in no way changes the fact that one can add them, and a GR solution remains, nonetheless, a GR solution. And in this sense, they are in no way forbidden. Only unobservable.
OP is about comparing what GR predicts for the case given - far field TT GW's owing to an oscillating linear mass quadrupole. Which nicely cuts to the base issue in a way obscured by the more popular orbiting binary treatments. I gave a reference for the field solutions. Which clearly confirms the illustrated local shear strains depicted in #1 have the correct orientations. The rest - impossibility of extending such purported local metric perturbations globally, follows immediately just by simple inspection of the axial symmetry involved.
Unfortunately, I'm unable to interpret this text in a meaningful way. Same for #19 and the "challenge" #8.
But, nonetheless, the situation does not seem as hopeless as one may think. The issue we have started to discuss seems highly relevant in all the interpretations of #1, #8, #19 I have tried, and this issue is a sufficiently clear and simple one, so that there is a chance to find some agreement.
But to improve this process, a few questions:
1.) Given that $\eta_{mn}$ is a trivial vacuum solution of GR, do you agree that the metric computed above from this using the coordinate transformation $x^m \to x^m + f^m(x,t)$ is also a vacuum solution of GR?
2.) Do you agree that above have curvature tensor 0?
3.) Do you agree that the $h_{mn}(x,t)$ computed above look (approximately) like waves, once the $f^m(x,t)$ are solutions of the wave equation $\square f^m(x,t) = 0$, thus, like gravitational waves?
4.) Do you agree that these four types of gravitational waves are the four types of waves which, according to the Living Review article, are not present in GR?
