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Flexible foam has shortest path from Point-A to Point-B. When the foam is not curved, the shortest path is Path-1 (straight line). But if the foam is curved, the shortest path is Path-2. If I think of the foam as space - should the shortest path (or geodesic) remain Path-1? Or is this analogy for space incorrect?

Flexible foam has shortest path from Point-A to Point-B. When the foam is not curved (space-time is not curved), the shortest path is Path-1 (straight line - before curving the foam). But if the foam is curved, the shortest path is Path-2. If I think of the foam as space - should the shortest path (or geodesic) remain Path-1? Or is this analogy for space incorrect? I think what I am trying to ask is that if in flat space, the shortest path is Path 1 (before the space is curved), if we add a large mass which causes the space around the mass to curve - as in the diagram, is the shortest path (geodesic) still Path 1? That is... does the scale for the metric tensor change along the inner radius of the curvature so that when we calculate the distance along Path 1 vs Path 2, Path 1 remains the shortest distance (because the metric tensor along Path 1 is different than the metric tensor along Path 2)?

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  • $\begingroup$ Path 2 is a geodesic. $\endgroup$
    – WillO
    Apr 22 at 0:39

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A better analogy is the surface of the Earth. Actually it's not an analogy, it's exactly the same math. The shortest path (OR longest path, i.e. extremal paths) between any two points on the surface is along a great circle. Yes you can bore through the Earth and find a shorter path, but not without escaping the surface and into a larger dimensional space around it. This mechanism is how a wormhole would work in 3D or 4D space (if wormholes turn out to be possible).

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  • $\begingroup$ So if the earth is deformed into a non spherical shape, do great circles remain geodesics? $\endgroup$
    – John R
    Apr 22 at 0:57
  • $\begingroup$ @JohnR. No. See en.wikipedia.org/wiki/Geodesics_on_an_ellipsoid $\endgroup$
    – BowlOfRed
    Apr 22 at 3:14
  • $\begingroup$ Thank you BowlOfRed! Maybe this is a slightly different question? Does a geodesic on a sphere remain a geodesic under some deformations of the sphere into a ellipsoid? $\endgroup$
    – John R
    Apr 22 at 3:32
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If you think that bending the foam changes the relative lengths of the two paths, then you are not thinking of the foam as space. If you were thinking of the foam as space, you would imagine a coordinate grid embedded in it; and when you bend the foam block, the grid would bend along with it.

As the grid bends, and as the two paths bend with it, the coordinates of the points along either path—foam-space coordinates—do not change. The foam-space lengths of the paths don't change. The foam-space shapes of the paths don't change. If path 1 was straight and path 2 was curved in foam-space before you bent the block, path 1 remains straight and path 2 remains curved.

Maybe more to the point: If the foam-space was flat before you bent the block, then it still is flat after you bend it. It doesn't matter that it no longer looks flat to your eyes. What matters in defining whether the space is curved or flat is the geometry as measured within the coordinate system.

In order to visualize a curved space, you have to visualize it as being "bent" in a higher dimension that is not contained within the space itself. E.g., as RC_23 said about the Earth. A 2D space defined by the surface of a sphere is curved in three dimensions. If you could visualize a curved 3D space (Hint: You can't) You would have to visualize it as being embedded in a higher-dimensional space—four or more dimensions.

See also: https://en.wikipedia.org/wiki/Manifold

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