## Friday, March 9, 2012

### Gravity Is Not A Rubber Sheet

Every physics demonstration of gravity uses the familiar “rubber sheet” model: We are shown a stretched piece of rubber, or perhaps the surface of a trampoline. A heavy ball is placed in the middle, distorting the sheet. Now a smaller ball, pushed in the general direction of the heavy ball, will follow a curved path, as if “attracted” by the mass. If given a particular kind of shove, it will circle around the heavy ball for a while, “orbiting” like a planet around a star. Thus the model demonstrates how an object with mass warps the fabric of space, causing the paths of other objects to curve in the direction of the larger object. Objects follow straight paths through space, but if that space happens to be curved by a massive object nearby, their paths will curve. Since Einstein, we’ve known that this is what causes gravitational attraction.

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.

1. The rubber sheet analogy is flawed from many perspectives, not the least of which is that it demonstrates neither Newtonian or Einsteinian gravity on a surface. It can't come near explaining a black hole and how space appears to start wrapping around itself as you approach the event horizon.

If you want to at least approximate Newtonian gravity, a particle's trajectory would be along a plane, and its path would be the plane's intersection with a cone for which the object that is attracting the particle would be at the point of tangency of a a sphere with the plane and which also fits into the cone where it is tangent along a circle. But that doesn't seem to be intuitive, and it doesn't predict the particle's velocity without further stipulations.

There isn't a nice way to present gravitation without math, so TV producers want a simple way to present ideas without forcing any undue thought processes. Sounds a little like religion, doesn't it?

Your comment editor is being pissy when I identify myself properly, so I'll have to use some option other than my real URL. This is getting ridiculous.

1. Thanks for the analysis. I wanted to write something about this, because someone seeing the demonstration might say, "Hey, that only works in gravity -- they don't know what they're talking about!" Recently, in a Stephen Hawking show on the Discovery Channel, they demonstrated gravity with a ship approaching a giant whirlpool. Of course, a massive object doesn't create a vortex, either. I appreciated seeing a different demonstration, but they never tell you that these visuals are crude analogies of the real thing.

2. I have also disliked the rubber sheet illustration of curved space motion, and am pleased by your exposition. I would however comment on two small points:
(1) The streched rubber sheet actually has NEGATIVE (Gaussian) curvature: consider any small piece of it, it is saddle-shaped. It's a consequence of it having minimal area. Therefore, geodesics do not converge,they diverge. So your two nudged balls would not approach, but separate.
(2) It may be experimentally simplest to consider rolling balls, but for the theory it's a terrible complication. A rolling ball, but with arbitrary spin, does not generally move along a straight line, even on a flat horizontal surface.
To eliminate the complications of rigid body motion, physics problems usually consider a block sliding without friction, but that's hard to realize. A better way would be a toy car with wheels set for straight ahead motion. If the wheels do not slip, such a car will move along a geodesic on ANY curved surface (radius of curvature >> size of car), EVEN if there is gravity!

brill@umd.edu

1. Thank you, Brill; that's very helpful. I've removed the reference I made to a "proper" rubber-sheet gravity demonstration in zero-G. (For the record, the essay previously ended with, "A true demonstration of 'rubber-sheet gravity,' perhaps in a zero-G environment — where rolling balls actually follow stable orbits around a surface due to geometry alone — would be interesting to watch. Until then, don’t take the terrestrial version too seriously."

3. The idea that the rubber-sheet analogy is supposed to describe is that objects move across geodesics in space-time. While that idea is true, I've never liked it, because it only really helps if you already know how to use Riemannian geometry.

A much better analogy would be this: in the presence of matter (or, more precisely, a nonzero stress-energy tensor), the causal structure of space-time warps, as represented by a rotation of the light cones for the object. The only problem with that analogy is that it first requires you to explain what a light cone is, but that's not really that difficult.

4. Today's xkcd shows us the actual use of the rubber sheet model:
http://xkcd.com/1158/

5. Nice post. Amazing information. Thanks for sharing. Rubber Sheet

6. One of the main problems I have with the rubber sheet model, is that two balls moving in the same direction with different speeds will take the same path, which is obviously not what happens at all in real life. Of course in relativity, you can't have the different speeds but the same direction, because you need to consider the time component of the 4-vectors. So then you get the problem that to really have any idea what's going on, you have to realise that the geodesic exists in four dimensional space-time and not just space. By this point it's getting so confusing that we might as well just pack away the rubber sheet anyway.