The content of this post is over simplified. It is not meant to be a definitive description of the involved subject(s). It is illustrative only and probably a flawed attempt even in that.
If there is no contact between mass and space ; what is there , in between mass and space ? ... What is the missing link ?
There is no missing link. I think that part of the problem is that you keep trying to think of space as if it were just another form of matter. If you look at matter it is itself mostly empty space. The nucleus of an atom of say lead is still about 26% empty space, between the protons and neutrons. Looking at the whole atom, including the electron "shell" around the nucleus and the atom has a radius about 10,000 times the size of the nucleus. And atoms even in solid objects, don't pack together such that they take up all of the space. Most of solid matter is actually empty space.
Think about neutrinos, only as an example! Depending on the source a neutrino is between 1/100,000th and 1/1,000,000th the size of an electron and has no charge or magnetic property. It does not seem even to interact at all with electrons. To a neutrino solid matter looks like empty space does to us. That is outer space. In fact astronauts are more likely to be hit by cosmic particles than a neutrino is likely to hit anything on it way through the earth. And a neutrino is still "matter".
Space has no more trouble passing right through matter than a neutrino would. In fact it most likely has less trouble than a neutrino does.
There is evidence that our Universe is expanding . Incidentally , this is also an example of physical movement of space . What stabilty problem is caused/observed , by the expansion of our universe . Our solar system is also part of the universe .
When physicists talk about the universe expanding, as in space itself expanding, they are talking about change that occurs cumulatively over great distances. Think about the "length" of one wavelength of light and the difference in the length of one wavelength of blue light compared to red light. The physical expansion seen in the cosmological
redshift represents an expansion of space equivalent to a few wavelengths of light at most and it occurs over cosmological distances, (billions of light years). Locally, here in our own solar system the expansion is so slow, that it becomes insignificant. It is just, to small and slow, locally.
You are correct in the statement that space "moves", but you cannot really think of that in the same way we think of matter moving. If we describe the interaction of neutrinos as a weak interaction with matter, (and in this description it is meant, not only that they seldom do interact, but also that when they do it is through the weak nuclear force only), we must think of the interaction of space and matter as very very weakly interacting.
If we consider this interaction from the context of the Lense-Thirring or frame-dragging effect, we find that space is drug along by the motion of matter moving through it, but..., think of that more like how how a flyswatter made of chicken wire affects the air it moves through, rather than a fan. And then scale that down many orders of magnitude, i.e. chicken wire would affect the air far more than matter affects space. Still they do interact. The exact mechanism we do not yet understand completely.
Perhaps physicists didnt make much study on space .
Physicists study space a great deal. The problem is that matter interacts so very very weakly with space that experiments to study that interaction are expensive and very difficult. (see the Gravity Probe B experiment and I think there is another either in the first stages or planned, by the EU)
The acceleration anomaly associated with satellites using a gravitational boost around a planet is one way it is thought that the frame-dragging effect of a planet may boost the satellites velocity. But this is not yet fully confirmed, as I mentioned the interaction is so weak that it is difficult to attribute its effect locally (within our solar system), to observations like this with any certainty.
I also agree with this that ; gravity is not caused by ' curvature(displacement) of space ' .
GR describes the interaction between space and matter as a curvature of space, which we experience as gravitation, this is something different than frame-dragging. There is no displacement of space that is detectable. And when dealing with planetary orbits even small influences should detectable, due to the scales involved. If there is any displacement at all, it is confined to the scale of individual atomic nuclei and likely the individual protons and neutrons of atomic nuclei, not even whole atoms, molecules or matter.
GR has been a very successful predictive theory of gravitation. There are no simple and clear examples to describe the gravitationally curved space of GR, in everyday terms. The closest is the bowling ball on a trampoline example and it is only vaguely similar. Perhaps creating more problems that it attempts to explain.
The main point I am trying to make here is that, we cannot think of objects displacing space. At best we might think of space being displaced by subatomic particles, but considering the scale, where a proton might represent between 1/10,000th and and 1/40,000th the size of an atom, that "possible" displacement would be insignificant and in the case of the gravitationally associated curvature of space, almost certainly not cumulative.
Do you mean to say that 'atomic-space' and ' outside-space' are continuous .
Yes. And that is without assuming that all of the space involved is dynamically locked. That it all move as a whole.
Think again about the frame-dragging effect. It falls off with distance, at least at the same rate as gravitation and perhaps even faster, (the even faster my own speculation). It can be expected that the angular momentum of protons, neutrons and atomic nuclei would produce a similar frame-dragging effect on space. However considering the scales involved, the frame-dragging effect of a proton cannot extend any significant distance beyond the particle or nucleus itself. If the earth were a proton and the moon were an electron the size of a hydrogen atom would be at least 120 times smaller than it is. The moon is about 82 times the earths radius from the earth, while a hydrogen atom is about 10,000 times the size of a proton.
The space directly associated with an atomic nucleus very likely has some dynamic interaction with the nucleus, that does not extend far beyond the nucleus. Keeping in mind the very very weak interaction, that does not mean it is separated in any way from space generally.
Space is space. It is dynamic on many scales. That dynamic aspect of space cannot be thought of, in the same way we think of atoms. At present we have no way to quantify space other than as a whole.
But this is not our experience of reality , that these two spaces are continuous .
It has to be. If they are not continuos there would be some displacement which we could measure or observer in how planets and objects in space interact. We do not see anything that suggests that space is displaced by the motion of matter.
Note: Remember, the examples and descriptions above have been simplifications and are not intended to be a definitive description.