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Comment by layoutIfNeeded | Hacker News Reader

How would periodic “free-fall” motion look in this setup? E.g. a point mass orbiting around a body, or a point mass oscillating back and forth in a 1D gravity well.

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This has some visualizations of space time curvature where you can get a sense of the lines particles take in different circumstances: http://www.relativitet.se/Webtheses/lic.pdf

I tried to explain it with words, but I guess the images are worth more than I could write..

In the left-hand image, an orbit would be a horizontal line, because it's a constant distance. So it's a mirror of the time axis, but translated upwards in space. It would be exactly the same axis-mirroring translation in the other images. So, importantly, it would not be a line in the right-hand image.

The concept of a constant distance orbit doesn't make much sense in 1D. A horizontal line in this model wouldn't indicate an orbit, but rather a completely stationary object. It would then make sense for the world line to curved because it must be accelerating in order to resist the attraction of the body it's near.

By definition, the world line of a stable orbit would be a line in curved spacetime.

With two space dimensions and one time dimension (2+1), an orbit in Newtonian physics is a helix. In general relativity that same helical path would be a straight line in an interestingly curved spacetime.

Likewise I guess for the 1D gravity well case, where a sine wave would become a straight line in spacetime.

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