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We have learnt that Waves of small amplitude cause particles in the propagating medium to do SHM ( simple harmonic motion ).

In the sheet of spacetime analogy ( I use this one because it is easy to put this down in words and visualize ). If gravitational Waves of amplitude small enough propagate through space time they should cause space and time to do SHM. what happens when this SHM happens in the opposite direction to the local gravitational field. Does the field strength change.

And what happens when local fields' strength is less than that of the wave. Do you get negative gravity. Or possibly true zero gravity at that point.

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I think this is a lovely question. In essence you are asking "when a gravity wave of sufficient amplitude passes through a region with very low gravitational field, can the effect of this wave be an effective gravitational repulsion with respect to the source of the wave"?

Let's think about this for a second. In a universe where there is just a single pair of black holes that orbit each other, you might be able to generate a gravity wave due to their relative motion. BUT - what you are really seeing in a gravity wave is a small perturbation of a larger gravitational field: in other words, when we describe a gravitational "wave", we are really saying that the gravitational field at a certain point is following the normal inverse square law, plus a small amount because the masses are moving relative to the observer.

I don't think that can ever lead to a negative value of the gravitational field (a "gravitational repulsion"). Intuitively (this is NOT a correct physical description, but perhaps a useful picture in your mind) if you think of a gravitational field as a stream of "gravity particles" traveling at the speed of light, then when the mass moves rapidly towards you and away from you, there will be some bunching and stretching of these particles; but they continue to travel towards you, and cannot "go the other way" (which would have to happen for gravitational repulsion).

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Firstly, it is a good question, do not understand why someone down voted without leaving pertinent comments.

As far as negative gravity, NO - Gravity is not negative, it may have different direction due to "negative phase". Once both the phases have passed, no net change should be caused.

As far as zero gravity - yes, it may be there momentarily provided "negative phase" exists (I do not know) and if it exactly balances the gravity that existed in absence of wave in magnitude and in direction for that moment. Free falling objects (including those in orbits) also experience zero gravity, so it is not impossible situation.

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