I just read this lecture by Feynman, where he describes curved two-D space as a “hot plate” where the temperature varies from place to place and it causes the yardsticks to change length and even bend. Usually I see curvature visualized as embedding diagrams. Are the two equivalent? What are pros and cons?
These are indeed the two main ways to visualize curvature, though few seem to be aware of the hot plate.
Embedding diagrams: Space(-time) is embedded in an external dimension. Examples: the 2d surface of a sphere in 3d, the standard “artist’s conception” of a black hole creating a “dip” in a grid of lines.
Deform in place: The “jello” of space(-time) deforms, and objects in it (particles, sticks…) deform along with the jello.
Alternatively, think of a 2d space being made of pixels (or a volume of space-time being made of voxels). These pixels measure 1 unit on the side. In flat space, they line up in a nice grid, in a plane. With curvature, we can no longer fit them into a grid. So they have to either keep their size and buckle out (embedding) or deform to no longer be square (hot plate).
This video compares the two visualizations in detail, and how you can switch between them:
And then the hot-plate version for 3d and 4d:
Pros and cons? It’s good to be able to think both ways. With an embedding dimension you allow for more topology, such as a wormhole connecting distant locations in one universe, or the “wrap-around” of a closed universe. On the other hand, IMHO, the deformations give more intuition, and help me analyze problems better.
Screenshot from video: