## Contours of impression

Here’s another weird one to exercise that graphics card sticking out of the back of your head. I was thinking about this while I tried to get back to sleep this morning. Consider it another flailing attempt at ideas beyond my ken.

The effect of gravity according to general relativity is often described as being similar to “a bowling ball lying in a rubber sheet”…

…which is probably one of the better mental illustrations of an abstraction ever described. But a number of problems occur to me, which may be illustrated by describing the actual, practical shape of such a rubber sheet:

At this point I should admit that I have very little understanding in this area, so this post is, in spirit, more of a question than a statement.

In this demo, the sheet will (ideally) meet the ball at a tangent point and then stretch linearly to the wood edges. If the weight is displaced uniformly across the sheet, then the stretching will be as well (I think?).

In one dimension the rubber sheet ought to be like a spring held between two fixed points. The bowling ball would then be balanced on the spring (in my head, I’m actually thinking of a heavy horseshoe hung over the spring, which seems less offensive), which would stretch the coils uniformly. I can’t think of any reason why the coils nearer to the ball/horseshoe should be farther apart than the coils near the fixed points.

If two bowling balls were in the same sheet, the contour of the sheet between them would be a straight line.

But in the idealization (top), we see that space is stretched more the closer it is to the point of origin. The contour of space between two planets in this model would not be a straight line.

The rubber sheet demo suggests a model where points in space prefer to stick together, like an endless chain-web of mechanical springs in two dimensions. The idealization suggests the same thing, with an added dynamic: the points in space also seem to prefer their unstretched position, creating a contour that returns more strongly to the unstretched plane. That’s more like a bowling ball on a Tempur-Pedic mattress.

Anyway, just musing a little. Curious why we think either dynamic (both spring and memory) applies to real space.

Also had some fun visualizing the effects of different-shaped heavy objects on the sheet as well, like a heavy ring or cylinder, or an object with a polygon footprint. Would recommend it.