How mathematically accurate is this popular classroom demonstration of gravity?

Question

I am referring to this demonstration: https://www.youtube.com/watch?v=MTY1Kje0yLg

The blue fabric is meant to represent space time, which becomes curved when the masses are placed, and then the curvature causes the masses to "attract" one another.

However, in our universe it is possible for a planet to orbit a star without ever colliding. It seems like in this model, the two masses will inevitably collide. This may be explained by energy loss through friction, but even if there were no energy loss from friction, would an orbit be possible? Also, if the fabric were infinitely big, would there exist an escape velocity or would any second mass always eventually collide with the first?

Our world's gravity can be well-approximated with Newton's inverse square law, but in this classroom demonstration, it is probably not an inverse-square law, so what kind of law is it?

Edit: In case the link stops working, here is a description of the demonstration. A large sheet of circular fabric is attached by its edges to a circular frame. When a ball is placed on the fabric, the Earth's gravity causes the ball to push the fabric down a little, stretching it - causing it to curve, which is meant to show the curvature of space-time. Now, when a second ball is placed on the side of the fabric, it will roll down the slope towards the first ball, which models gravitational attraction. Other balls can be placed on the mat with an initial speed such that it "orbits" around the first mass in the middle, but the orbiting masses always end up colliding with the first mass.

Answer 1

This is a terrible model, in my opinion, and even worse, 39 million people have viewed it. A computer simulation is a much better idea.

This may be explained by energy loss through friction, but even if there were no energy loss from friction, would an orbit be possible?

You ask this because you will never get that (stupid) slope idea out of your mind.

Orbits are obviously possible, but if you take the formation of the solar system as an example, although there is no complete picture of how exactly the planets wound up where they are now, you can forget the slope part as a force causing the planets to "fall" down. They formed instead from a collection of smaller rocks that was already in orbit in roughly a flat plane around the equator of the Sun.

Wikipedia: Protoplanetary Disc

Also, if the fabric were infinitely big, would there exist an escape velocity or would any second mass always eventually collide with the first?

Again, the model is to blame for this question. If a planet, or more likely a comet or asteroid, approaches an existing solar, the chances of it moving into a stable circular orbit are remote. How this planet interacts with the Sun falls into one of three categories, assuming it is not aimed directly at the Sun.

If it is moving fast enough, it can come close to the Sun and then escape with a higher speed than it came in with, which is what you mean by escape velocity. Comets can sometimes do this, and we use the same technique for space probe velocity boosts.

If it is not moving fast enough, it will enter a chaotic orbit, (the only part of the model that makes any sense to me) and eventually it will hit the Sun.

The least likely option is that it will have just the right velocity vector, both in magnitude and direction, to enter a stable orbit. I am guessing this is extremely unlikely without the "just right" intervention of the gravitional influence of another planet(s).

Answer 2

If your only purpose is to use an analogy where curvature causes movement, I guess it works as that: an analogy. But students must understand that an analogy is not the same as an explanation, and it can't substitute for one.

The most common criticism of the rubber sheet is that it depends on an external force to work, and that of course is valid. But there's more: this analogy misses many important aspects of the curvature of spacetime, particularly one that's right there in the name: spacetime. The fact that time flows differently at different places is crucial here. To give an example of why, if you take the gravitational field near a spherical object (like the Earth) and remove the factor that makes time "flow differently" at different heights, then you no longer have gravity: it's possible for an object to simply stay in place wherever you place.

I must say that I don't know if the rubber sheet reproduces an inverse square law, but I don't think it matters very much. If your question is how mathematically accurate the demonstration is, then I think the answer is clear: not at all.