My understanding of relativity explains that the presence of mass warps space-time so that light travelling through the warp follows at straight line but the warp itself is curved and therefore the light seems to an outside observer (non-local reference frame) to be travelling in an arc.
If the mass is sufficiently large, such as a planet-sized body, the curvature will be centred around the centre of mass (the core) and radiate out in all directions. Like a pinch at the centre of a soft foam block.
Relativity points out that at different heights from the Earth's surface time moves at different speeds, the closer to the core the slower time gets. In the case of a singularity time slows considerably as an observer gets closer to the centre of the curvature. Dimensions (x,y,z) also get tighter near a singularity.
As space-time is in effect the one entity, space and time itself gets "smaller" as an observer gets closer to the centre of a large mass.
Can this shrinkage constitute a compressibility of space-time?
UPDATE:
I have obviously over simplified and while it hasn't helped others, I can't apologise as it is my thought process to start simple and slowly add new data into the thought experiment until something closer to reality is generated. Let me explain my thought process a little better in the hope of generating more discussion. Please continue to be kind as the thought experiment is a stage, not real life.
What do we know:
As an object moves it interacts with space-time. As the object's velocity increases, so does its mass, exponentially. This becomes the barrier which states no object can exceed the speed of light. Maybe this is due to the sum constituent particles of the mass interacting with various space-time quantum energy fields, notably the Higgs Field, or just plain frame dragging per Relativity. Time dilates and mass increases, adjusting as C remains a constant. Space-time is warped around the object. Is this warp symmetrical like a star, or more like a 3D bow wave? If the object slows to a small fraction of C then the warp around it dissipates and returns to its resting entropy. Is space-time's "quantum foam" is warpable?
Now take two very high mass objects, neutron stars and black holes. Find 2 of them, in any combination, close to each other and they will be orbiting each other at near relativistic speeds. These generate high levels of frame dragging that interact with each other to generate gravity waves. If they were much further apart there would be no gravity waves, just local warping.
Before they were neutron stars or black holes they were stellar objects of mass greater than the Chandrasekhar Limit. They were much larger and far less dense. During the cataclysm that created their current state they blew off significant mass but still ended up much denser. So I posit a question: The density of these objects change as they transform, what does this do to their gravity field? A second question: How does the density:gravity ratio of the object alter during the transformation?
Have we as humans measured the density:gravity ratio of stellar objects before and after such events? Should the increased density after the transformation increase the gravity levels at the same point from the centre of the gravitational field than before, I would suggest that space-time is compressible.
Going back to one comment that space-time is naturally warped, can we believe that both natural warping from the events of previous interacting gravity waves and relativistic stellar objects (ejected from the orbit of a black hole, etc.), as well as the presence of matter, warp space-time? Both seem to fit. Space-time could be perceived as a dynamic topology constantly changing from natural forces. Areas of high density and low density interact with each other, pushing and pulling based on their "polarity", according to gravitational laws? Food for thought I hope.
Best Answer
If we consider the thought experiment where we take the classic 2 dimensional plane curved in a graphic representation of the curvature of space-time, copy it and arrange the copy so the lowest points of the gravity wells are aligned. These copies can be arranged any way you like, as long as the low point, or gravity wells are aligned, then the imagery still works. Then we can see what i think you are asking about. With just a single image, it appears space-time can be stretched and pulled 'downward.' But in 3 dimensions, the second image, or many others, are also possible, implying that space is being curved down and up in the same place.
And so, i assume, your question. My thought was, where did the space go?
The other things that i consider are distance and time. Using the earth for example, over time the earth moves around the sun. In the summer the earth is curving different space-time than it does in the winter. As the earth leaves a place in space, space returns to the shape it was before the earth was present, so it seems that space-time was 'compressed' and then returned to it's original shape.
I find it interesting that space can be expanded, inflated or curved but not compressed. The confusion comes from the comparison of space to a fluid, and fluids cannot generally be compressed. Keep in mind that my first example, the rubber tarp example, and comparing space to a fluid all give us a idea of what space is like and what it is doing, but none of them are perfect.