Black hole collision animation: What are these extrusions? In this video uploaded by LIGO Lab Caltech, two inspiraling black holes are depicted. The video's description explains what is shown and can be summarized by: 



  The colored surface is the space of our universe, as viewed from a hypothetical, flat, higher-dimensional universe, in which our own universe is embedded.


  ... the colors depict the rate at which time flows.


  [Space] is dragged into motion by the orbital movement of the black holes, and by their gravity and by their spins. This motion of space is depicted by silver arrows...



Just before things become quiescent, regions of space around the merging black holes extrude upward. (Actually, the extrusions seem to begin sometime before things get chaotic.) If "space" here is being represented like the rubber sheet analogy, what do these extrusions mean?

If gravitational forces create depressions in the sheet, then it seems to follow that those extrusions are anti-gravitational, which can't be right. Could they be regions where the equations modeling the black holes' interactions spit out nonsense?


(From video, timestamp 0:52)
The shape of the surface shown in the video is a depiction of the spacial curvature of the spacetime. (The relationship with time are depicted seperately by the arrows and the colors.) More particularly, the shape is depicting the curvature of equatorial plane of the binary. The depicted surface has been embedded in a (fictional) 3D space in such away that the curvature of the surface is equal to the intrinsic curvature of the equatorial plane.

Lets try to unpack what this means for the interpretation of the extrusions. First note, that there is no physical meaning to whether something is show as an extrusion or a depression, this does not actually effect the curvature. The video makers could have also chosen to depict the depressions around the black hole as extrusions instead --- without changing the meaning.

What is relevant, however, is that some regions are shown as depressions while others are shown as extrusions. This means that somewhere in between, there must be a saddle point in the depicted surface. Saddle points correspond to regions with negative (spatial) curvature (i.e. an area where if you would draw a triangle its angles would sum up to less than 180 degrees). The extrusions themselves quite clear have positive curvature (a triangle would have more than 180 degrees).

Note that the sign of the spatial curvature has little to do with whether gravity is attractive or repulsive. If you want an indication of direction in which gravity is working, the arrows give a better idea (although that interpretation should also be taken with a pinch of relativistic salt).

EDIT:
Clarifying the last point a bit. The animation depicts three aspects of the spacetime curvature: The rate at which time flows (the lapse) as a color map, the rate at which space is dragged (the shift) as gray/silver arrows, and the spatial curvature as the curvature of the surface. Together these three completely characterize the curvature of spacetime. Consequently, they dictate how a test object would move in the spacetime, i.e. "how gravity acts". Although all three elements are important for the motion of particles, some give a better qualitative indication of the behavior of test particles than others. In this respect, the color map and the arrows are more important than the spatial curvature. The typically a particle will want to move with the arrows, and along the gradient of color towards the redder regions (in both cases this generally means toward the black holes). The spatial curvature plays a somewhat secondary role, and matters mostly for particles moving at high velocities. Hence my comment that the sign of the spatial curvature is not a good indicator for whether gravity is attractive or repulsive at some point.