4 ms·

The "balls on a rubber sheet" is a pain because nothing is in free-fall: there are dissipative contact forces between the balls and the rubber sheet. Consequen

The "balls on a rubber sheet" is a pain because nothing is in free-fall: there are dissipative contact forces between the balls and the rubber sheet. Consequently realistic initial [position, velocity] values for the test ball cannot give you a stable circular orbit around the central mass ball. Venus isn't about to fall into the sun. Now try setting up Earth-Moon or the Jovian-Gallilean systems on the rubber sheet. It's fun to try to peel away defects in the "ball on a rubber sheet" in an effort to arrive at Newtonian limit equations of motion for balls in placed around it. (First advanced question: how do we adapt Einstein-Hilbert to reflect trapping onto the sheet? Does dimensional reduction work?) What sort of membrane in constant acceleration could generate scaled 2+1 timelike geodesics for model solar system objects placed on it? Can one scale this to a model of the solar system with all orbits flattened onto the membrane? For instance, what do the IVs look like for a ~ 1050:1 mass ratio between a model sun and a model Jupiter that is shrunk down to classroom size and retains a good match to Jupiter's real orbital parameters (or if you prefer, lengths and angles) across many orbits? Without destroying this scale model orbit, can we add the inner solar system? Can we get sensible orbits of scale model Galilean moons? (And of course all of the above has completely neglected the rotations of these bodies about their axes, which is clearly always very wrong with a typical in-classroom rubber sheet + balls demonstration. We can blame friction for that.) I'm not sure what you're representing on silly putty: are the drawn lines solutions to geodesic equations? What would the twists-pulls of the putty in a scale model ~kg:g Sun and Jupiter system look like? Or are you thinking about a relativistic regime somewhere in the right half of this diagram : <https://en.wikipedia.org/wiki/Post-Newtonian_expansion#/media/File:GR2bodyparameterspace.png https://en.wikipedia.org/wiki/Post-Newtonian_expansion#/medi...> ?

cyberax 2y ago
> 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.
- oneshtein 2y ago
  Same level relative to what? This model of gravity requires gravity to work.
  - cyberax 2y ago
    To be precise, it needs a force that would bend the rubber sheet. You can use a spring to do that if you want.
- raattgift 2y ago
  > concept of geodesics 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.
portn0y 2y ago
Don’t call it “balls on a rubber sheet” then. Describe it as an artistic representation of the theorized behavior. Pull a Maxwell, whose theory of electromagnetism only worked when he got rid of the imaginary levers; get rid of descriptions in terms of physical things. As a visual it’s fine. The debate here is the language. Only one aspect needs to change.

cyberax 2y ago

> 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.

oneshtein 2y ago
Same level relative to what? This model of gravity requires gravity to work.
- cyberax 2y ago
  To be precise, it needs a force that would bend the rubber sheet. You can use a spring to do that if you want.
raattgift 2y ago
> concept of geodesics 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.

portn0y 2y ago

Don’t call it “balls on a rubber sheet” then. Describe it as an artistic representation of the theorized behavior. Pull a Maxwell, whose theory of electromagnetism only worked when he got rid of the imaginary levers; get rid of descriptions in terms of physical things. As a visual it’s fine. The debate here is the language. Only one aspect needs to change.