General Relativity part 2: Curvature in 3 and 4 dimensions.

In six Level 1 videos we visualize General Relativity with a substratum (“fabric”) that uses the same math as GR, the math of curved surfaces.

Here in part 2 of 6:  We move on from 2d space to 3d space and 4d spacetime. In the Schwarzschild spacetime, we find that space contracts in the radial direction, and time slows down near a black hole. For a shell, we see that the flat spacetime inside the hole differs from flat spacetime far away.

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4 Responses to General Relativity part 2: Curvature in 3 and 4 dimensions.

  1. Michael McGinnis says:

    The voxels inside the spherical shell should be compressed slightly more than those outside the shell. But, will they still have a square cross section? Obviously the space dilation is greater inside the shell than outside the shell but will it be isotropic?

    • bernander says:

      Time dilation increases as you move deeper in a spherically symmetric mass, but generally not radial contraction. I base this conclusion on MTW eq. 23.27’ (note the prime on the equation number): for a spherically symmetric mass, the radial part of the line element is dr^2/(1-2m(r)/r), where m(r) is the total mass inside radius r. For a shell, m(r) is zero inside the shell, so the compression vanishes, just as it does for large distances. If you know of a contradictory result, please let me know, and point me to the source.
      Inside the shell, space is flat, but with clocks running slower than far away.

      • Michael McGinnis says:

        Space is flat in a spherical shell but it’s still dilated relative to space far from any mass. Space and time dilation are always proportional or c would change. At the center of gravity of a massive object, spacetime dilation is maximal.

        • bernander says:

          I appreciate your dedication to this issue, though maybe you have in mind different definitions of “stretch” than I do.

          In my visualization I sprinkle spacetime with rods that locally measure 1 unit. I have no compression of rods along the circumference (except for the rotating black hole and spinning ring). In a visualization, rings that differ in measured C by a dC are rendered as if their visualized radii, Rvis, differ by dRvis = dC/2pi. (Rvis is what you get if you put a ruler on the computer screen while watching the video.)

          Then in a Schwarzschild spacetime radial rods compress (not stretch), so that we measure more r than suggested from the measured circumferences C1 and C2 of neighboring rings; Feynman calls this “excess radius.” A local observer (with his local, compressed rods) thus measures more than Rvis.
          But inside a spherical shell, where space is flat, the observer measures exactly Rvis: no compression.

          If the visualization inside were to have compressed radial rods, it would need to stretch the circumferential rods, and the center point would end up being another spherical shell with the manifold not defined inside that inner shell. I would find that visualization confusing.

          So, as visualized, the videos should have correct radial size for voxels inside the shell.

          You mention a “time stretch” and a “space stretch” and a “spacetime stretch,” and I don’t know what you mean. Stretch relative to what? In my videos, there is a time dilation near a mass RELATIVE TO far away, and you can measure it with clocks that are initially synchronized, let them sit at the two locations for a long time, then compare total measured time. I also have a measured radial *compression* RELATIVE TO measured circumference.

          And I don’t know from where you get the rule “Space and time dilation are always proportional or c would change.” Again, dilation relative to what? Maybe you have another visualization in mind where your various stretches makes sense (bulging into an embedding dimension?), but that’s not the visualization in my videos. If you are developing an alternative visualization, I have a friendly suggestion that you rig it with unit rods, and that you make sure the rigging is consistent with the metrics (Schwarzschild, flat, etc.). My rigging (where voxels are made from 12 unit rods) should be.

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