What does this have to do with the rigidity of space? Think of a location that is about 40 miles from where you are at this moment. Even if there is interstate highway between you and this location, it would still take quite a drive to get there - 40 miles is a fair distance. Now, imagine that the entire region around you is covered with a layer of insulating foam, which is 50 feet thick (think of something you might find in the cushions of a sofa).
If you were to set a golf ball on the foam in front of you, it would have no reason to roll in any particular direction (for this example, assume that you are somehow hovering above the foam, and that there is nothing else nearby to disrupt its smooth, flat surface).
Now, let's say that 40 miles away there is an enormous floating crane, which is holding a scale model of the sun (a spheroid, 36 feet in diameter) suspended in the air. A sphere this size would have a volume of roughly 24,429 cubic feet.
To stay somewhat true to the model, let's say that the sphere is filled with liquid hydrogen. Even though hydrogen is much lighter than, say, water, a sphere this size would still weigh just under 54 tons on earth.
Now, if the crane were to begin slowly lowering the model sun into the foam, it would sink completely to the bottom and become enveloped within it. It is also safe to say that the depression created by the model sun, which would have the appearance of a gravity well, would not impact our golf ball (40 miles away) in the least. In fact, from this distance there would be no way to know that anything had happened at all. All of the impacts of the presence of the model sun would be absorbed by the surrounding foam, and probably would not produce any observable effects reaching more than a few dozen yards.
Clearly, foam is too soft a medium to transfer the effects of the deformation caused by the model sun for very far at all.
What if the region around us were overlaid with an enormous, perfectly flat steel plate rather than foam (that is, following the curvature of the earth's surface). This would certainly change the analogy some. Even then, it is unlikely that even the highest grade, densest steel could convey the impacts of the resulting depression far enough to reach the golf ball and cause it to roll. We would be closer to doing so, however, since the steel would probably be more than dense enough to transmit the vibrations caused by the event for 40 miles.
Still, if we imagine the plate of steel being 80 miles in diameter, into which we forcefully create a depression of 18' (half the diameter of the model sun), could we expect the golf ball (even one without dimples) to roll towards it? Even if the steel plate were only a few feet thick rather than fifty, it is not likely that the golf ball would roll at all. The steel would bend and absorb the impacts of the depression well before they reached 40 miles - they would simply dissipate.
This leaves us wondering: What material could be so strong that a depression of only 18' in its center could cause a golf ball, 40 miles away, to roll towards it? Surprisingly, there is something.
It is estimated that the spacetime fabric is a billion billion billion times stiffer than steel (Ripples on a Cosmic Sea, David Blair and Geoff McNamara). That's a big number (1043). In the coming posts, we will begin to see that this rigidity is more than only another fascinating cosmic detail. It is a fundamental, pervasive and highly influential factor in the behavior of the Universe, having implications that can help explain some of its more difficult-to-understand characteristics, and move us closer to understanding certain mysteries that only come to light when we appreciate its influence.
Note: Many things in physics have no direct correlation to what we might think of as common reality. Analogies of bowls, rubber sheets, foam and trampolines are rather crude instruments of communication in any discussion about things as abstract as Dark Matter and spacetime.
We should only know that the goal of this discussion is not to painstakingly create perfect illustrations, or to exhaustively point out the flaws in less-than-perfect ones. It is, of course, about the riddle of Dark Matter. These illustrations are only disposable artifacts, created and destroyed along the path to what I hope to be a greater insight.