A wave has peaks and valleys. I can think of a sine wave as a wave with peaks and valleys. Now, if gravity is a wave, can we say that gravity would have peaks and valleys, with the valley becoming negative? In other words, in the picture of gravitational waves, can gravity become negative or 0? If yes, what does negative gravity mean? Repulsive gravity?
Gravitational waves are waves of tidal forces. If a gravitational plane wave passes, it alternates pulling you apart in one direction and squashing you in the orthogonal direction.
First of all, waves can have any shape. Sines and cosines are the simplest waveforms, and in fact you treat a wave with a more complicated shape as a sum of multiple sine and cosine waves with different amplitudes and frequencies called a Fourier series. This fact is helpful, because it means you can figure out how any wave behaves by considering only the simplest waves.
Before we think about a gravitational wave, let's think about a sound wave. The thing that oscillates in a sound wave is the pressure. As the sound wave passes, the pressure increases above ambient pressure, and then decreases below ambient pressure. The equilibrium point of the oscillation is not zero pressure, it is the ambient pressure. The pressure oscillates up and down, but it doesn't ever go to zero or to a negative value.
A gravitational wave has similar behavior. The thing that oscillates in a gravitational wave is the metric. The metric tells us how to measure distances in spacetime and is related to the curvature of the spacetime. But the there is always an ambient, non-zero, equilibrium metric. When the metric increases the spacetime between objects expands, increasing the proper distance between them. When the metric decreases the spacetime between objects shrinks, decreasing the proper distance. The oscillation doesn't make the distance go all the way to zero or flip the sign of the distance to negative.
A wave has peaks and valleys.
This is not necessarily true. In particular, it assumes the existence of an order relation on the values of the wave (i.e. that we can meaningfully say that the peaks are "greater" than the valleys). However, this is only the case for scalar quantities like in sound waves (pressure) or water waves (height). For other types of waves, like electromagnetic waves (electric field and magnetic field vectors), it does not make sense to talk about peaks and valleys (unless dealing with some scalar quantity calculated from the vector/tensor/etc). Since gravitational waves are oscillations in the metric tensor (i.e. not a scalar), it is not particularly meaningful to say that they have peaks and valleys, or to say whether they are positive or negative.
Waves are moving perturbations of and/or on a substantial medium. So, if "space" (or "space-time") is either nothingness or an immaterial fabric of spooky geometry (or pure 4-D maths), then how could gravity make waves?
BTW, "space" is an artifact of our perception of "things" and "places" that exist in our "field" of experience. Of course, all of that is enabled by the intrinsic principles of being, like form, structure, functionality, dimensionality, causality, identity, multiplicity, complexity, etc. "Time" is a concept we associate with our misperceptions of change, etc. Thus, a "fabric" of curvy "space-time" is clearly as fictional as unsupportable beliefs about "dark" matter/energy that we cannot directly detect. Hence, no amount of beautiful maths will ever make fictions, conjectures, approximations, and maps more real than the territory (the cosmos).
Yet, if somebody can explain how an omnidirectional "force" (G) without a motivator can cause effects (waves) in nothing, then I will be amazed. However, I will then expect to see their proof of the Navier-Stokes equations problem and the Yang-Mills band gap problem and get the two $1-million prizes (before I do). So, best of luck etc. ~ M