5

I understand that gravity is not a force, but instead that mass/energy curves space around them and hence, a object near some considerable mass, though always is travelling in a straight line in it's frame, it is actually following a curved trajectory because of the induced curvature (when vied from some external reference frame).

I've come across different visualizations of curved space, and I'm trying to intuitively grasp what they represent:

  1. Straight up curved, circular lines around a central mass.
    This makes sense to me as a simplified $2 \text D$ representation, but I'm unsure how this extends into a fully $3 \text D$ visualization of spacetime curvature.

First curved space depiction

  1. The second style shows a curved grid around a massive object, giving a more direct sense of the spacetime fabric being distorted.
    While this helps convey the idea of "space being curved," I find it misleading in some ways. It shows trajectories where objects fall into the central mass but doesn’t account for stable orbits, and others where the object passes on by deflected from its original path.

Curved space depiction two

Are there more effective ways — visual, conceptual, or otherwise — to develop an intuitive understanding of curved spacetime that:

  • Explain why free-falling objects are actually following curved paths
  • Account for both infall and stable orbital trajectories
  • Better represent the 4D nature of spacetime curvature?

Any recommendations for improved models, simulations, analogies, or visualizations would be very welcome.

This question came to me when I myself was trying to simulate curved spacetime. I came up with the following, though I am not really sure about what I've made.
I went with the second style (though I didn't add the flow), which shows similar curvature as the object moves around in $2 \text D$.

Self made simulation

cconsta1
  • 1,257
Kyathallous
  • 1,006

1 Answers1

6

Are there more effective ways — visual, conceptual, or otherwise — to develop an intuitive understanding of curved spacetime that explain why free-falling objects are actually following curved paths?

That first gif actually isn’t terrible at visualizing this. It’s not great, but it works. Locally, spacetime is flat, so you as an observer can always draw a coordinate grid around yourself and use that to measure things very close to you and at rest w.r.t. you. But this is just a local property of spacetime, and globally, the grid might twist or curve around. Locally, you always see yourself going “forward” along one gridline or another, and you perceive these gridlines as locally straight, even though they may be globally curved.

Any recommendations for improved models, simulations, analogies, or visualizations would be very welcome.

The best way to interpret spacetime curvature is as the change in the magnitudes and directions of the basis vectors $e_\mu$ of the coordinate system being used to chart whatever set of points makes up the manifold (of course, this is not intuitively helpful).

For the purposes of drawing, none of the cited three are really correct. A good visualization, but not actually a great representation of what’s going on. They are all very gravity-centric: things will go from out here, wherever that is, to in here, wherever that is. And they are good at representing that. But this is also easily describable with just a bunch of arrows pointing to “in here”.

For most purposes, this will be sufficient. Either 1) you need enough detail to describe the actual spacetime and should use equations/graphs rather than animatios, e.g. technical work/presentations, or 2) you don’t need much detail and can model whatever you want on the backend and put whatever looks cool up front.

If I were asked what the best possible way to represent the curvature would be, in three dimensions at most without needing colors or other such fancies to represent the fourth dimension, I would plot a grid of light cones (at least, the first bit of them). One thing all spacetime curvature does, gravitational or cosmological-expansion or frame-dragging, is change the shape of light cones; if that could be visualized, with a 2D grid with light-cones at each vertex facing “upwards” away from the grid, I think that would probably be a faithful representation.

If you really need a 3D grid I think colors would be permissible to label the future vs. present part of the light cones: you would have a 3D grid, and at each vertex place a ball representing the segment of the light cone, with the outermost “furthest-future” parts colored red and the innermost “closest-to-present” parts colored blue.

controlgroup
  • 8,801