Does a massive object contract space-time or expand it? Before asking, I must say that I am not a physicist of any sort, but I do have a strong interest in Relativity that has led me to question my layman’s understanding. So please stay with me on this “thought experiment” despite my lack of formality.
At center is the so-called rubber sheet analogy for how massive objects such as the Earth warp space-time. Similar to how a bowling ball is placed onto a trampoline, the surface will warp and anything on a straight path that crosses this warped surface will have its trajectory altered by the curvature of space-time. Light does this when it curves around the gravity of the sun.
Given a few things I’ve heard, I wonder if the expansion of space-time by massive objects is actually true. Is the space warped outwardly because of a “bowling ball” on a tense surface, or does the “bowing ball” actually make space-time more dense? I ask because there are three things I’ve considered. I will cover two of them first.
Considering it factual that:


*

*An object free-falling toward a massive body without resistance will accelerate proportionally given its distance.

*The closer one is to a massive object, the slower time moves.
My thoughts are that the reason an object in free fall will accelerate toward something more massive, is because it crosses to regions of evermore condensed space-time.
My third thought is that less time passes but more “distance” is covered.
That said, I also considered how a gravitational wave might interact with a gravity well. If as Einstein said was true, then all gravity should propagate equal to the speed of light. However, if one considers that a gravity wave both contracts and expands space-time as it propagates, then is this not a wave that may potentially move faster than light in its locality?
My suggestion here is that massive bodies gravitationally attract space instead of expanding it, and this eventually could be seen in the data with the likes of LIGO and future experiments. I’m not betting my life on it, but I’d like to offer a suggestion to people way smarter than me.
Does mass contract or expand space time?
 A: You need to be careful about statements like spacetime is contracting or expanding or indeed doing anything else. Spacetime isn't a thing. It is a mathematical object that we use to describe the motion of things. So it is meaningless to ask whether spacetime contracts.
However what we can do is take a sphere of test particles and see how it changes as it moves through spacetime.
Imagine taking a large number of particles that are too small to exert any significant gravitational force on each other and arranging them in a sphere. If these particles are floating in space far from any other matter then they will just stay as a sphere - the radius and volume of the sphere won't change. But if we now let our sphere move into some gravitational field then it will change in shape and/or volume. So while it doesn't make sense to ask if spacetime expands or contracts it does make sense to ask if our sphere expands or contracts and how that sphere changes does tell us about the curvature of the spacetime.
It turns out that the volume of the sphere is related to an important properties of the spacetime geometry called the Ricci tensor and Ricci scalar aka scalar curvature. Basically an increasingly positive Ricci scalar means the sphere is contracting and an increasing negative Ricci scalar means the sphere is expanding.
The shape of the sphere is related to another property called the Weyl tensor. This tells about tidal forces acting on our sphere.
To make this concrete take your example of the sphere falling towards a massive body - possibly a black hole or possibly just any massive object. When we calculate the spacetime curvature we find that the Ricci tensor and scalar are both zero so the ball stays the same volume. It neither contracts nor expands. However the Weyl tensor is not zero so the ball experiences tidal forces. In fact the ball gets stretched along the line towards the massive object and compressed at right angles to this line. This process is what s commonly known as spaghettification.
So for a massive object if we were going ignore the imprecision and use the metaphor of spacetime spacetime expanding or contracting we would have to say that spacetime is neither expanding nor contracting near the massive object, but it is changing shape.
There are examples where the volume of our sphere does change and the most obvious is an expanding universe. In an expanding universe like ours we find the volume of the sphere increases with time, which is why we talk about the universe expanding. In a hypothetical contracting universe we would find the volume of the sphere decreases with time. 
A: I love this type of question. Sounds like an obvious question in the first second and then it really isn’t. I’d assume the bulk of students would not dare to raise a hand and ask this question for fear of looking silly; yet I’d bet most not being sure about the answer.
And neither am I. Here is my attempt:
If you approximate the universe as a combo of empty space plus point-like masses (Isaac Newton says hi), which isn’t too shabby an approximation, then space is, at each point, a saddle surface, so the absolute amount of curvature is zero everywhere (as in being positive in one direction and negative in the other). While this is not at all evidence that spacetime would not contact or expand around a body, to me it helps to imagine that only curvature, not density of space time, is of relevance.
Thought experiment: If I’d make spacetime globally twice as dense right now (it’s a wednesday at 2:30am right now, perfect time for some shady stuff); how would an observer notice? If that would shrink distances, velocities, wavelength and everything else uniformly, it won’t have any effect to anyone within the system. So space density would be a moot point, unlike curvature. (Sort of like absolute motion is not a thing, whereas acceleration and rotation is).
If you knew the curvature of every point of a time-slice of spacetime, you could probably integrate local space-time density differences based on curvature. My understanding is that then, close to masses, spacetime is more dense*; but this “density” is meaningless to anyone within the universe; even including two observers observing one another from regions with two different densities. The curvature between these areas, however, is observable.

*) or only space gets denser close to a mass at the expense of time density? my brain hurts. help.
A: 
Does mass contract or expand space time?

No.

My suggestion here is that massive bodies gravitationally attract space instead of expanding it,

No, not that either.
Mass causes curvature of spacetime in general relativity.
