Does mass compress space-time? 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.
 A: 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.
A: The presence of matter doesn't warp spacetime. You could make a shell of matter and the spacetime on one side could be perfectly flat no matter how close to the shell you get.
Spacetime warps naturally. A wave of warped spacetime could propagate through a universe that is everywhere devoid of matter and always was and always will be. It's natural for spacetime to be warped.
What matter, or more correctly, what stress, pressure, momentum density, energy density, and energy flux do is allow regions of naturally curved spacetime that normally could touch meet up. For many systems the energy density is the largest factor and the energy density is almost proportional to the mass density.
The way spacetime outside a star managed to get as curved as it did is because there was some small curvature earlier and some matter. And the matter was slow enough relative to each other that the small amount of curvature allowed it to compress. And as it compressed the type of curvature of the outside now couldn't change to a different type like it did back when the matter was there.
So the type of the outside had to extend inwards and it was a type that has more curvature closer in. And the gas collapsed more and this process continued.
Eventually when all the matter that makes our sun collapsed to be as small as the orbit of the earth, the spacetime here was as curved as it is now. The gas contibued to collapse so that type of curvature extends all the way to the surface of the sun, whereas it used to be the type that was outside the whole solar system and the curvature was weaker there but the same type as what we know have here.
After that you seem fine. It's not the presence of matter here and how that matters, it's the curvature here and now that matters. And curvature exists and propagates itself and the matter influenced it when it passed through a long time ago.

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.

This is a simplification that went too far. There isn't a global time that ticks at different rates in different places. Instead clocks move through space and time and when they tick is a function of where they are and how they are moving.
I bring this up because of the next mistake you are about to make.

In the case of a singularity time slows considerably as an observer gets closer to the centre of the curvature. 

Nope. A naked singularity doesn't do that. And a singularity behind an event horizon will have an infinite time dilation relative to infinity at the horizon, not at the singularity. So you are confusing a horizon and a singularity.
A: General Relativistic curvature is "intrinsic curvature" - that is it doesn't require the existence of another dimension "to curve in". The warping/curving is in a direction that can be component-ised into our familiar x,y,z,t.
(Another way in which the age old analogy of a ball rolling a warped 2D sheet is flawed. Unless the sheet was warped in its own plane, that would be better.)
Such intrinsic curvature, if different at different locations, inevitably will result in higher and lower "density" of space time. Which is to say that identical objects on parallel free fall paths will appear to converge or diverge when the encounter such curvature. They will appear to have been acted on by a force.
We already know the concept of gravitational potential energy (the metric tensor in GR). For a reference mass/energy this could be interpreted as measure of the "density" of space time, with differentials representing areas of higher curvature, and higher apparent gravitational force. 
But it would be wrong to suggest the gravitational potential (not the rate of change of gravitational potential) is proportional to the curvature/force.
A: Catalyzer beware! There is very little to be saved in the answers I
have read, preceding mine. At least in part, it also depends on how you expressed your question. I would have much difficulty to give you a correct and understandable answer.
These are not matters one can explain with ordinary language. Whoever tells you the contrary, maybe did not understand himself - surely he is bound to make you misunderstand deeply.
I offer you an ancient quotation:
"Philosophy [i.e., physics] is written in this grand book - I mean the
universe - which stands continually open to our gaze, but it cannot be
understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering around in a dark labyrinth."
Galileo Galilei, Il Saggiatore (The Assayer, 1623).
A: 
does mass compress space-time

There is not an answer to this question because it is folly... Space-time isn't a real thing, space and time are two different things entirely, Space is that which can be occupied by matter and energy and time is the speed at which two matters and or energies or more move in relation to each other.
For example there is no such thing as a past or a future, there is only present time, and yes you have a memory but that is only recordings of what present time once was. Time is only the change of present time and the change of objects in present time compared to where they were, once in present time... 
However space can compress matter, at the center of every world that has gravity, there is a black hole, all different sizes depending on many things, and what a black hole is, is the point in which space is collapsing. Space is a different thing than energy and matter. Matter is compounded energy, and energy is postulated particles in space.
Space can collapse and it can be expanded endlessly, because matter and energy has to exist inside of space and it has to have its own space. "Two objects or energies cannot occupy the same space" because it has to exist in a space it sticks to the space it is in unless forced out by another energy. So when that space is being pulled toward a black hole, the matter travels with it i.e Gravity. If it was the first piece of matter to be "pulled" into the black hole it would sit at the center, because matter will not collapse as space does, so a new piece of matter comes toward the center of the black hole and runs into the other matter there, now you have matter trying to occupy the same space, so there is force and heat is generated. 100 million years later or so you have a planet.
When there are two black holes both collapsing space into their center, the space and an even distance between the two is stretched in both directions creating a "force" which keeps them apart. And so a never ending cycle of collapsing and stretching space is born between two worlds.
When a planet has collected so much matter that it is at the point where the mass outreaches its collapsing space and is extending into the space than is stretching you have massive lifting and collapsing of matter, where whole continents would separate from the bulk of the planet and then slam back into it over and over to the point where no stability is obtainable and the entire planet ignites into molten material and a sun is born.
A sun cannot die because of 3 basic laws:


*

*Energy cannot be destroyed it can only be altered.

*Two objects cannot occupy the same space.

*All energy and matter must occupy a space.


So the sun will survive until the day that whatever created the black hole at its center is remedied. At which point it will blow up, which the people of earth are calling supernovas... or whatever.
A: All matter is moving through space-time. The bigger the mass, the slower time appears. All matter is moving from where time is fastest to where time is slowest, where time is at its slowest or stops completely is evidenced by a "black hole". Warping occurs because of the interaction of matter with the space-time. Gravity is simply matter moving to where time is slowest.
