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I've been wracking my brain trying to understand what "curved spacetime" really is, and I think replacing one dimension with the time dimension then drawing the world-lines through time was the "aha!" moment.

And this obviously goes ahead and explains our relative perception of gravity appearing to be a constant force downwards, whereas we really just following a curved world line through time (which we are always moving along) that is curved downward toward the Earth.

I added a satellite orbiting to show another aspect, that is that the space station could be said to be constantly moving "up" away from the center of mass, but the curvature keeps it at the same distance to the Earth. We can't see that extra dimension so we feel like we are "falling".

Is this diagram of the world-lines through time correct? The orbiting space station is just an extra thought experiment I put in.

(click image for full size)

Space through time

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    $\begingroup$ I don't understand what you mean by "the actual feeling of spacetime curvature." $\endgroup$ – Alfred Centauri Mar 1 '15 at 2:30
  • $\begingroup$ What space-time curvature feels like is gravity. Really. That's the day-to-day experience that you have that corresponds to curved space-time. $\endgroup$ – dmckee Mar 1 '15 at 3:00
  • $\begingroup$ @AlfredCentauri by getting a feeling of what the curvature really means using our generally 3-dimensional thinking brains. Once I realised the diagram above, it all made sense, is what I'm saying. $\endgroup$ – Nick Bedford Mar 1 '15 at 3:02
  • $\begingroup$ In other news, it's bizarre to walk around knowing that forward is really "downward" a little. $\endgroup$ – Nick Bedford Mar 1 '15 at 3:03
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    $\begingroup$ Nick, I don't believe it's likely that your drawing shows the actual feeling of spacetime curvature. Great circles on a sphere, where latitude is the time coordinate and longitude is the space coordinate, do a much better job in my opinion. See, for example, physics.stackexchange.com/a/79683/9887 $\endgroup$ – Alfred Centauri Mar 1 '15 at 3:23
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No.

You seem to be implying that the fact that the world-line of the satellite looks curvy, is what is meant by "curvature of space-time". No, that's wrong.

The world-line of the satellite looks curvy in the picture, yes. But, it should be clear that you can have a curvy-looking world-line without gravity. Anything that orbits anything for any reason--e.g., a cowboy whirling a lariat--would have a similarly looking world-line.

Anything that is subject to any net force (not necessarily caused by masses acting gravitationally) will have a partially curvy looking world-line.

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  • $\begingroup$ How would you propose drawing a 4-d rendering in 2-D? $\endgroup$ – user33995 Mar 1 '15 at 3:20
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    $\begingroup$ I wouldn't. And how it that relevant? $\endgroup$ – hft Mar 1 '15 at 3:25
  • $\begingroup$ Granted this is artistic interpretation. but the "curvy line" as he explains it would be a 2d+1 representation of a 3d+1 event on a 2d surface. $\endgroup$ – user33995 Mar 1 '15 at 3:47
  • $\begingroup$ But it does not have anything to do with what we mean when we say "spacetime is curved by massive objects", which--I believe--is what the OP thinks he is explaining with the picture. $\endgroup$ – hft Mar 1 '15 at 3:59
  • $\begingroup$ Well, the orbit is only there show the relative path of the orbiting satellite with the large body relative to curved spacetime. The curvey orbit line is just curvey because it's an orbit... I.e with respect to "space", the satellite is constantly moving away and sideways from the surface of the Earth (if that's correct). $\endgroup$ – Nick Bedford Mar 1 '15 at 4:43

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