Can we "stir" space-time? Depending on the energy-momentum density of a given region in space there is an intrinsic curvature to it. I can't help but feel like space-time is a like fluid, which begs the question whether it is possible for the points "on" our space-time manifold to be displaced? I not talking about gravitational waves, I am asking whether we can create holes in space-time or displace points on space-time such that a given region has a higher density of "things" which space-time is made of?
Note: I think I could not explain myself as clearly as I wanted so I would appreciate any help for making the question less convoluted.
 A: In classical General Relativity, the thing we call spacetime has two layers of mathematical structure. The underlying layer is a smooth manifold, which has just enough structure to define things like continuity and derivatives. By itself, a smooth manifold has no notion of distance or time. The overlying layer is a metric field. The metric field defines distance and time, and it is the mediator of the gravitational interaction. 
Regarding the metric field as a "fluid" — in the sense that it can have higher density in some places and lower than others — it problematic because the metric field itself is what defines things like volume, so it is a prerequisite for defining the densities of other things.
On the other hand, the metric field is indeed a dynamic entity. In both influences and is influenced by whatever else occupies the spacetime, such as the electromagnetic field, material objects, and so on. The frame-dragging effect, already mentioned in niels nielsen's comment before I wrote this answer, is an example of this.
Roughly speaking, the Einstein field equation describes the influence in one direction (how everything else influences the metric field), and other equations of motion describe the influence in the opposite direction (how the metric field influences everything else).
