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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 (From video, timestamp 0:52)

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  • $\begingroup$ I don't think the extrusions represent anti-gravity; they represent regions where the local gravitational acceleration is much less because they are "between" the two objects (if you went out to infinity, the level of the plane would be higher than the extrusions). $\endgroup$
    – antlersoft
    Nov 19 '19 at 16:36
  • $\begingroup$ @antlersoft If you read into the description further than my summary, it says that the green regions are where (my paraphrasing) "time is not so much affected," (compared to a distant observer) so space-time curvature isn't so drastically distorted in these regions. I thought that might be the case too before I read the video description. $\endgroup$
    – BMF
    Nov 19 '19 at 16:40
  • $\begingroup$ I remember seeing this animation as well. I always thought that illustrated gravitational waves from both objects colliding with each other. Think about swirling both of your hands around each other in a pool of water. Each hand creates a wake, and when the two wakes collide the energy from each pushes the water even further up, above the crest of each wake individually. I'm not sure if that is a direct correlation to gravitational waves, but this is how I understood it. $\endgroup$
    – Greg Burghardt
    Nov 19 '19 at 18:12
  • $\begingroup$ @GregBurghardt If the model is using the rubber sheet analogy, then those extrusions would be analogous to anti-gravity, unless it isn't using a rubber sheet analogy, although by the video description it seems like it is. $\endgroup$
    – BMF
    Nov 19 '19 at 18:18
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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.

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    $\begingroup$ Thank you for your answer! What I don't understand is your last statement, that the sign of spatial curvature has little to do with whether or not gravity works attractively. (I should note, most of what I understand about physics was gleaned from popular science.) I thought negative curvature was a characteristic of an anti-gravity. (For instance, the positive curvature of mass-energy in the universe is seemingly balanced by dark energy, a negative curvature influence.) I'll look into this more because I'm lacking an understanding, but could you also elaborate a bit on that? Thanks again! $\endgroup$
    – BMF
    Nov 26 '19 at 14:50
  • $\begingroup$ Gravity doesn't induce space-time curvature, curvature describes gravity. So, I think your last statement needs to be elaborated, although, I'm starting to think that maybe the extrusions are an artistic choice by the animators ... However, the length of the arrows across the extrusions are clearly longer than those in positively-curved space, implying something like negative curvature. $\endgroup$
    – BMF
    Nov 26 '19 at 22:46
  • $\begingroup$ Hmm, okay, after your edit I see my mistake in interpreting the arrows, although, I'm still unsure how you explain the redder regions. Are the depressions and extrusions both representing positive curvature in some way? Why would it be that the gravities of the black holes are shown as depressions while curvature of the binary plane is shown as extrusions? $\endgroup$
    – BMF
    Nov 28 '19 at 14:58
  • $\begingroup$ @BMFForMonica Because the direction perpendicular to the plane does not have any physical meaning. It is the shape (curvature!) of the surface that represents the spatial curvature. Areas that are locally curved like a sphere represent positive curvature, while the saddle points represent areas with negative curvature. It is the existence of these saddle points that require the positive curved area to extend in different directions from the surface. $\endgroup$
    – mmeent
    Nov 28 '19 at 15:50
  • $\begingroup$ It is further worth noting that the way that the curvature is divide in the 3-different aspects shown, strongly depends on the chosen coordinates. This is something to continually keep in mind when interpreting visualizations like this. $\endgroup$
    – mmeent
    Nov 28 '19 at 15:57

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