> The "balls on a rubber sheet" is a pain because nothing is in free-fall
The balls on a rubber sheet model is actually really great, but not in the way it's typically presented (rolling a ball down the curvature). Instead, just use a pen to draw lines to show the concept of geodesics.
Start like this:
1. Imagine that you can move without friction if you stay at the same vertical level.
2. Draw a line on a flat rubber sheet, that's a line in free space. It's just a straight line that can go on to infinity.
3. Now put a ball onto the rubber sheet, so you get some curvature. Now the lines near the ball that stay on the same level are not straight lines, but circles.
Ok, you brought in the "g" word. I think that makes it fair to ask how you plan to calculate the analog of spacetime intervals along your rubber sheet.
The general curves (not just geodesics) down to the heavy central ball all end up where the ball and sheet are in contact unless blocked by earlier contact with the central ball. How do you recover anything like the behaviour of a smaller ball bound to a non-spacelike radial geodesic with one end normal to the central ball's surface and the other "at infinity" (or at least in the part of the sheet that is asymptotically parallel to the ground)?
How do we show students, using your adapted rubber sheet, that spacetime curvature around a spherically symmetrical mass is symmetrical in all the spatial dimensions and that there is a curvature induced on the timelike dimension too.
What's the relationship between your pen lines and the Christoffel symbols (notably \Gamma^{1}_{00}) or Euler-Lagrange?
How does your "3." work with highly eccentric elliptical orbits? How about a grazing hyperbolic orbit? Think Halley's comet and 1l/`Oumuamua.
The balls on a rubber sheet model is actually really great, but not in the way it's typically presented (rolling a ball down the curvature). Instead, just use a pen to draw lines to show the concept of geodesics.
Start like this:
1. Imagine that you can move without friction if you stay at the same vertical level.
2. Draw a line on a flat rubber sheet, that's a line in free space. It's just a straight line that can go on to infinity.
3. Now put a ball onto the rubber sheet, so you get some curvature. Now the lines near the ball that stay on the same level are not straight lines, but circles.