When I was first getting interested in physics, the rubber-sheet model of gravity bothered me. For one thing, it only works in gravity! It seemed that the rolling ball was just curving downhill. Tilt the sheet without warping it, and its path will curve the same way. In the weightlessness of the International Space Station, I figured, the model wouldn’t do anything. I didn’t like that gravity was required in order to demonstrate how gravity works. It was like a model that shows where wind comes from, but which only works when it’s windy.
Something else disturbed me. When the rubber-sheet model is presented in diagram form (in books, for example), the diagrams are often inconsistent. Empty space is depicted as a flat grid of straight lines, but when a massive object is added, some of the lines suddenly form circles. The graph-paper grid turns into a pushed-in dartboard or spider-web pattern, with circular elements representing potential orbits around the mass. Thinking that maybe I had discovered something, I wondered: At what point do the open-ended straight lines of empty space start joining together to form closed circles? If we took an empty region of space and gradually started adding mass to it, when would the circles appear? I was perplexed — the diagrams never show that transition, just the before and after!
What’s wrong with this picture?The problem of course lies not in Einstein’s theory, but in the rubber-sheet model. It isn’t a perfect analogy for gravity.
It’s a coincidence that real gravity on Earth causes a rubber sheet to warp in a manner that suggests the warping of space. You could just as easily turn the model upside down, and push the ball up against the rubber sheet, and the sheet would be warped in the same way (just in the opposite direction). The rubber-sheet model of gravity is intended to demonstrate how a massive object causes space to curve, so it’s the warping of the sheet that’s important, not the direction.
When a two-dimensional surface is curved into a third dimension, its geometry changes. No longer do the laws of Euclid, which most of us learned in 9th grade, apply: The angles of a triangle do not add up to 90°, for example. In ordinary geometry, two parallel lines never meet; in the non-Euclidean geometry of a curved surface, parallel lines can meet. Imagine that you and a friend began walking from the equator to the north pole. Initially, your paths would be exactly parallel, but since the Earth’s surface curves, the paths would intersect at your destination. Similarly, if two objects were moving in parallel from empty space toward a star, their paths would eventually converge — even with no sideways forces acting upon them.
As it happens, the rubber-sheet model would work in zero gravity, if you warped the sheet with some other force (say, by pushing the end of a broomstick against it), and if you got the rolling ball to remain on the surface somehow (perhaps with a bit of static electricity). In that case, the ball’s path would appear to curve as it attempted to follow a straight line on this non-flat surface. And two balls, nudged along parallel paths toward the depression, would approach each other as the surface under them began to curve.
As for the grid that’s often laid over the rubber sheet, it’s only there to help you see the shape of the surface. The straight or circular lines are a human invention; there is no such grid in space. The actual paths that objects trace through warped space are, well, the actual paths that they trace. These can be circles, ellipses, parabolas, or hyperbolas, depending on the trajectory of the object.
Planets and comets go their own way — they have no use for grid lines.The rubber-sheet model does give a general idea of how gravity deflects the path of an object. But it’s a crude demonstration, as the Earth’s gravity fouls the geometric effect that the model is intended to demonstrate.* When you see the rolling ball get “attracted” to the larger ball, much of that deflection is just the ball rolling downhill, as it would on a tilted, flat surface. A true tabletop demonstration of gravity, where objects follow stable orbits along a surface due to geometry alone — would be interesting to watch. Until then, don’t take the conventional version too seriously.
* Consider what would happen if you rolled a ball inside the surface of a vertical tube in a frictionless vacuum. Under Earth’s gravity, the ball would inevitably spiral down to the floor. But in zero G, it would follow a circular path forever. This circular orbit, not the spiral, is the accurate representation of the “straight-line path” that would be followed on the surface due only to its geometry.
