What is actually waving in a gravitational wave if spacetime is not a thing (just a mathematical construct)? I have read this question:

except what is waving is spacetime itself.

Gravitational Wave - What is waving?
Is gravitational wave a new category of wave?
Yet none of the answers are satisfactory, because on this site you can read when somebody asks whether spacetime itself is a thing, that spacetime itself is not a thing, and we just measure spacetime distances between events in spacetime.

In light propagation, oscillation does not mean any movement in space. It is the value of the electromagnetic field, at one given point in space, that oscillates.

How does light oscillate?
Now the example of electromagnetic waves gives you on this site a nice example of how the electric and magnetic field components intensity is oscillating, and that in reality when you visualize EM waves, these field components intensity is what "waves". Still, the electric and magnetic field components intensity "waves" or oscillates in spacetime. So for the EM waves, there is always a higher "level" of something they are enclosed into, that is spacetime hosts them.

However spacetime is not a thing - it is a mathematical construction. Specifically it is a manifold equipped with a metric. At the risk of over-simplifying, a manifold is a thing that has dimensionality (four dimensions for spacetime) and a metric is a function that defines distances between points in the spacetime.

How is Space-Time produced?
But what about gravitational waves? There is no higher level something they would be enclosed into, nothing that hosts them. Spacetime itself is waving is not an answer (maybe it is just not obvious), because even on this site, certain answers suggest that spacetime itself is not a thing.
So basically what I am asking is, if spacetime is a mathematical construct, a human creation, then what is waving in a gravitational wave (that can carry energy). Naively thinking, a mathematical construct cannot wave and carry energy.
Question:

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*What is actually waving in a gravitational wave if spacetime is not a thing (just a mathematical construct)?

 A: This is quite an insightful question, in particular the explicit distinction with an electromagnetic wave propagating in a fixed background spacetime.
There are two points I would make.

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*Spacetime geometry is a useful abstraction for scales where quantum gravity is not important (which is the regime of every experiment done to date and very likely well into the distant future). Rather than saying whether a given concept, like the electromagnetic field or the wavefunction or the spacetime metric is real or not, I prefer to frame physics concepts in terms of whether they are useful. For example -- is pressure real? You could argue that it is just an emergent, mathematical property of a gas, because if you zoom into microscopic detail you see that pressure is really some average force being imparted by $10^{23}$ discrete particles banging into the wall of a container. However, I think it's much more useful to say that on the scale where we are describing what happens when we squeeze a balloon, that the pressure of the balloon is real. In an analogous way, in essentially all situations of interest, it is incredibly useful to think of spacetime as real. Even if we never directly measure the spacetime metric, we can infer it by performing a set of measurements of distances between points, and the time intervals it takes light to travel along certain paths -- which is essentially what LIGO does. If we are dealing with quantum gravity, things become more subtle and it's not clear the concept of "spacetime" is useful anymore -- but we are not obligated to think about quantum gravity when dealing with macroscopic objects, and in fact it probably isn't useful to do so, much like it is often better to describe experiments using a balloon in terms of pressure instead of directly in terms of atoms.


*Gravitational waves are small perturbations in a background spacetime. Mathematically, when dealing with gravitational waves in "ordinary" circumstances (not traveling over cosmological distances and not very close to a black hole), we write the metric $g_{\mu\nu}$ as a sum of a known, flat background metric $\eta_{\mu\nu}$ and a small perturbation $h_{\mu\nu}$. (This can be generalized to more general background metrics; I'm only ignoring cosmology and black holes for simplicity.) The fact that there is a fixed, background spacetime, gives something for the perturbation $h_{\mu\nu}$ to propagate in, much as you pointed out we often describe the electromagnetic field as propagating in a fixed background spacetime. This approximation enables us to do things like define a local stress-energy tensor carried by the gravitational wave, which is not possible to do exactly in GR. You are right, that when dealing with a general, time-dependent spacetime, where the "gravitational waves" are just as important as the background -- for example, very close to the collision of two black holes -- GR is extremely complicated and non-linear, and it's not usually possible to cleanly separate "a wave" from "changes in the background". In practice, the way to deal with this, at least for black holes which are a lump of matter and curvature inside a flat spacetime -- is to look at the asymptotic spacetime, which is approximately flat. Then you can define gravitational waves as small perturbations on the flat background. You can also formalize this idea to define geometric quantities like the Bondi mass which capture the same information.
Therefore, my answer to your question is: spacetime is real (or at least "it's so useful to think about spacetime as if it were real that it's pointless to quibble about it" on scales of interest) and gravitational waves are small oscillations in a fixed, background geometry. For situations where the metric is extremely complicated and time-dependent, you need to use some trick like looking at the metric at asymptotic infinity to be able to separate gravitational waves from the evolution of the time-dependent background metric.
A: What is waving is the metric tensor in my understanding. The spacetime is there as a manifold, but its metric oscillates and that is the gravitational wave.
For example, suppose a perfectly static and round planet. The Schwartschild metric results in a symmetry for lengths North-South and East-West in a point of its surface. The passage of a gravitational wave can make that lengths oscillate, being no longer exactly the same. The Schwartschild equation is modified to include a dependence with time, and not only with the radius.
A: What is actually waving in a gravitational wave if spacetime is not a thing (just a mathematical construct)?
Notice, no one ask what is an electromagnetic wave because the field is real and the radiation associated with the field is real (i.e light).  Gravity is also real and it has a real field.  If spacetime is truly describing gravity then the gravitational field is waving.  Right?  No.
Remember,  the same question was asked about the De Broglie wave when it was first used in connection with electron orbits.  The question about what was waving was never really answered and the debate continued until realist like Einstein, Schrodinger, and DeBroglie were dead and then the Copenhagen Interpretation for modern physics prevailed.   Which just means we have inherited an unreal way of describing and looking at the world.  Nothing can really be known and there is no objective reality, just multiple universes where anything goes.
Your question is less about gravitational waves and more about the state of theoretical physics when it is divorced from mechanics and engineering and then wedded to the magical world of abstract mathematics.
Einstein quote "Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore"
A: What is actually waving in a gravitational wave if spacetime is not a thing, just a mathematical construct?
Spacetime, as described by General Relativity Theory, is a mathematical construct. Therefore, gravitational waves are just a solution of the corresponding differential equations. That we are able to register them with our artificial senses means that Einstein's theory is a good mathematical model for this physical phenomenon (gravitation). However, it does not exclude the existence of spacetime as a thing. Some other conclusions from Einstein's theory suggest that spacetime have some physical properties like physical bodies. For example, the divergence of central pressure for the critical $M/R$ ratio in Schwarzschild interior solution strongly reminds of stress singularity at crack tip in solid body, https://www.researchgate.net/publication/331944629_On_stress_singularity_at_crack_tip_in_elasticity, or G.W. Gibbons's conjecture of Maximum Tension Principle, https://arxiv.org/abs/hep-th/0210109 , which makes statement about the constant force $c^4/G$, seems to have  a deep physical meaning. Mathematics's “world“ is dimensionless.
A: There is a speculative new theory that postulates that all time dilation is due to an excitation of the gluon field, which in turn leads to a dilation of the particulate space of loop quantum gravity.  It is the size of these particles that determines the speed of light, and therefore the speed of time.  When two black holes collide they generate a wave in the gluon field, analogous to a wave in the electromagnetic filed. It is this wave, traveling across space, that we consider to be a gravitational wave.
As I said, this is speculative so don't quote it in any paper, but it will give you something to think about.
Now the details:
As you know, all of space is filled with an electromagnetic field and a gluon field.  The electromagnetic field is "excited" when near a bar magnet or by a flash of light. The gluon field is excited when near a planet filled with spinning gluons. It can also be excited by a galaxy moving through the gluon field at high speed which generates a bow wave, or by the collision of two black holes.  Both of these fields can be considered as "liquid media" so they are easily disturbed.  If two black holes collide (as discovered at LIGO), then it would create a big ripple in the gluon field.  This ripple of excitation in the gluon field will run across space at c, in exactly the same way as a flash of light runs across the electromagnetic field at c.  As the ripple passes each point in space, the energy it contains will lead to a swelling of particulate space causing a momentary dilation of time, and thus is a gravitational wave.
Spacetime is the combination of the size of particulate space and the time that size imparts.  A warping of spacetime, the basis of modern gravity, is essentially an excitation of the gluon field, leading to a dilation of particulate space and a dilation of time.
Very speculative, but it does provide a complete explanation for the physical process of gravity and gravitational waves.  I have yet to find anything wrong with the theory, but it does rely on Loop Quantum Gravity being true.  You can find the paper on this here
