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How to resolve confusions on the rubber sheet analogy of the spacetime curvatures?

I am a newbie to spacetime curvature. I have watched several youtube videos on Einsteins GR and spacetime curvature where most people used the analogy of the rubber sheet and ball on the sheet. However, I cannot realize that rubber sheet analogy with my reality. For example,

  1. The rubber sheet is a two dimensional sheet on the other hand I see our universe is a three dimensional.
  2. The balls on the rubber sheet are placed from one direction to other (top to bottom) but in reality I see out planets and stars are flying on the space.
  3. Light flows over the rubber sheets and curves around the balls in its world line, but in reality we know everything themselves are travelling with the speed of light in their world line.

How can I resolve these confusions? I am really interested to know detail about them.

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    $\begingroup$ Analogies are intended to be similar to something; they are not the same thing. So they are always imperfect and partial. Not every aspect of the analogy will be reflected in reality. The rubber sheet analogy has many serious problems, as several earlier questions on this site show. $\endgroup$ – Anders Sandberg Nov 10 '20 at 21:39
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    $\begingroup$ I can't understand the English in your third point. Please clarify. $\endgroup$ – James K Nov 10 '20 at 21:40
  • $\begingroup$ @JamesK I tried to edit 3rd point $\endgroup$ – Sazzad Hissain Khan Nov 10 '20 at 21:52
  • $\begingroup$ Finally, I got the answers from this fantastic video youtube.com/watch?v=wrwgIjBUYVc $\endgroup$ – Sazzad Hissain Khan Nov 11 '20 at 0:13
  • $\begingroup$ Think of poppy seeds in rising bread dough. The farther apart any two seeds are, the faster they are moving apart. $\endgroup$ – Wayfaring Stranger yesterday
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The rubber sheet only is not meant to be a qualitative model, it gives one concept and one concept only: Mass causes curvature of spacetime.

You can't get any more than that from the rubber sheet. If you have that idea in your head already then you are ready to drop the image because:

  • The sheet is 2d but spacetime is 4d
  • The 2d sheet is embedded in 3d space. Space-time isn't embedded in a 5d or higher dimension (or at least, if it is, it is irrelevant)
  • The sheet has two space-like directions with no time dimension. Space-time has a time-like dimension.
  • The basic way of finding distances on a flat sheet is $(x^2+y^2)^\frac12$. The basic way of finding distances in 4d spacetime is $(x^2+y^2+z^2-t^2)^\frac12$ (where units are chosen to make the speed of light = 1, eg time in seconds, distance in light-seconds)

You can't create a simple image of curved 4-d space-time. The rubber sheet analogy will hinder further understanding if you try to use it to understand why an object will travel in an apparently curved path in space when in a gravitational field. It's only purpose is to embed the notion that "space-time is curved" it can't tell you anything else about what that implies.

So to address your three points

  • In fact spacetime is 4 dimensional.
  • Yes. The placing of balls is strictly metaphorical. There is no actual sheet.
  • Object don't move at the speed of light. In their own frame of reference they are still (by defintion). If you have two objects, they can move relative to each other, but always at less than the speed of light.
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    $\begingroup$ Regarding the 3rd point, objects which stay still travel in 100% of its time direction (which is light speed) from past to future. Please correct me if I am wrong. If not wrong then why another light travel paths are usually shown across the curvature around the balls in the rubber sheet analogy to instill the concept of attraction towards each others? $\endgroup$ – Sazzad Hissain Khan Nov 10 '20 at 23:42
  • $\begingroup$ You can't use the rubber sheet analogy to understand why an object or light appears to follow a curved path. In GR, objects in gravitational fields are straight (geodesics) in curved 4d spacetime. A moving particle (point) in space makes a line or curve in space time. If the particle is not acted on by a non-gravitational force, its line in spacetime will be a geodesic (ie straight) But when you take 3d slices of spacetime, and make a movie, you don't see a line, but a particle that moves and it moves in an apparently curved path. $\endgroup$ – James K Nov 11 '20 at 9:23
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I just wrote an answer about this.

The rubber sheet is a good model of 2D Newtonian gravity, with a 1/r force law. If you make a rigid surface in the shape of the 3D gravitational potential, like the gravity wells you find in some science museums, and roll small balls on it, it's a pretty good model of orbits in 3D Newtonian gravity, with the correct 1/r2 force. The surface is still only 2D, but 3D orbits always lie in a plane anyway, unless more than two bodies are involved.

Curved surfaces with balls rolling on them, whether rubber or rigid, are not a correct model of general relativity. General relativity is something like an ant crawling on a curved surface. A crucial difference between a rolling ball and an ant is that if you turn a gravity well upside down, the ball will roll away from the center, but the ant will still follow the same path as before. The ant doesn't care about the background gravitational field. However the ant's path on a typical GR embedding diagram is not a correct model of GR either, because as you noted, the ant should really be traveling through time, and the embedding diagrams usually don't include the time direction. The ant's path on these diagrams is a path that a tachyon might take, but not any actual particle.

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