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I was wondering-how do you visualise curvature in the context of general relativity. The gravity well and trampoline analogies are quite wrong, so I want a more realistic approach to it (say, the way Einstein himself might have visualised it). Mathematically, it all makes sense, but I am not really sure how does this really looks like. More specifically:

  1. How do you visualise Riemann tensor?
  2. Ricci tensor?
  3. Weyl Tensor?

ANY DIFFERENCE IN THE VISUALISATIONS?

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    $\begingroup$ Can you see in six dimensions? I think that's the least number of dimensions a "realistic" embedding would take. The way Einstein visualized it was with good math skills and an enormous physical intuition. It was not with graphics. $\endgroup$ – CuriousOne Dec 30 '14 at 15:06
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    $\begingroup$ Related: physics.stackexchange.com/q/2447/2451 , physics.stackexchange.com/q/21065/2451 , physics.stackexchange.com/q/92741/2451 and links therein. $\endgroup$ – Qmechanic Dec 30 '14 at 15:14
  • $\begingroup$ @CuriousOne: "Can you see in six dimensions?" I should be able to if they exist. Because, why shouldn't I? $\endgroup$ – bright magus Dec 30 '14 at 15:44
  • $\begingroup$ @brightmagus: Evolution in 3d. $\endgroup$ – CuriousOne Dec 30 '14 at 15:47
  • $\begingroup$ @CuriousOne: Not true. $\endgroup$ – bright magus Dec 30 '14 at 15:57
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No actual diagrams, but this paper looks like it could be helpful: Physical and Geometric Interpretations of the Riemann Tensor, Ricci Tensor, and Scalar Curvature. And there's also this course outline on John Baez's site, and Visualizing Spacetime Curvature via Frame-Drag Vortexes and Tidal Tendexes which does have a bunch of diagrams.

Also, you say "The gravity well and trampoline analogies are quite wrong"--it's true that the "rubber sheet diagrams" you often see cannot really be thought of as "gravity wells", but they can be defined in such a way as to accurately depict proper distances in a 2D subsection of a curved 3D hypersurface of simultaneity from a larger 4D spacetime, see my answer here. It's also possible to similarly "embed" a 1+1 dimensional cross-section (with one spacelike dimension and one timelike) of a larger 3+1 curved spacetime in a 2+1 flat (Minkowski) spacetime, see here.

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  • $\begingroup$ The problem with the embedding into the Minkowski spacetime is that we can't visualize that, either. Just embedding Minkowski itself doubles the required dimensions, already. It basically takes a higher dimensional saddle shaped manifold just to show special relativity correctly. $\endgroup$ – CuriousOne Dec 30 '14 at 16:44
  • $\begingroup$ @CuriousOne - We can't perfectly visualize Minkowski spacetime since the apparent visual length of timelike/spacelike curves doesn't correspond to their proper time/proper distance, but Minkowski diagrams do have some intuitive features, like the fact that straight lines in a Minkowski diagram are always geodesics, and that light rays always travel at 45 degrees from the time axis. And since the 1+1 curved spacetime surface is being embedded in a 2+1 Minkowski space, not a 3+1 one, it doesn't double the required dimensions, just adds one (so we can have a 3D diagram with one axis as time). $\endgroup$ – Hypnosifl Dec 30 '14 at 16:58
  • $\begingroup$ I agree about Minkowski diagrams, but we are basically stuck in shoestring land. I have not seen a proper diagram even for flatland. The only book I ever saw that had details about the math of minimally required embeddings gave some insane numbers like 2n+1 for the general case. $\endgroup$ – CuriousOne Dec 30 '14 at 17:10
  • $\begingroup$ @CuriousOne - Yes, for the general case you need a large number of dimensions to embed in a flat spacetime, a paper referenced here apparently was only able to find a lower bound of 90 dimensions with up to 3 timelike ones. But the paper in my linked answer about spacetime embeddings said on p. 16 that the radial plane of any "'normal' static and spherically symmetric star-like object (without horizons or regions of infinite density)" can be embedded in 2+1 flat spacetime. $\endgroup$ – Hypnosifl Dec 30 '14 at 17:25
  • $\begingroup$ It is interesting that one can do at least a highly symmetric (if unphysical) case "nicely". I will definitely add that to my collection of "exceptions to the rule that nature in general is nasty". Thanks! $\endgroup$ – CuriousOne Dec 30 '14 at 19:09

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