In General Relativity, gravity is described as the curvature of spacetime caused by mass. This curvature is often visualized as a straight path bending due to a warped surface. My question is: If, hypothetically, spacetime had a linear instead of a radial curvature (think of a constant pull in one direction), would an object's motion be affected differently compared to the standard radial case?
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$\begingroup$ By the equivalence principle, this would be equivalent to flat spacetime as seen a uniformly accelerated observer, i.e. spacetime in Rindler coordinates. $\endgroup$– Michael SeifertCommented Mar 3 at 15:24
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$\begingroup$ related: physics.stackexchange.com/a/777842/24093 (it depends on how you construct your uniform gravitational field, there are multiple ways to do it in GR but none of them are physically achievable) $\endgroup$– YukterezCommented Mar 4 at 23:26
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The uniform field in one direction is basically what we experience on the surface of Earth.
The illustrations you see like the rubber sheet bent by a heavy object are just trying to give you a rough conceptual idea. They are not a correct description of General Relativity when you analyze it in detail.
The thing to remember is, space is not just curved – spacetime is curved. And in weak gravity fields, nearly all the curvature is in time, not in space.
Here is a good explanation that is scientifically accurate, with good visualizations as well. The channel itself is a great guide for introduction to moderately advanced physics concepts
First let's take a familiar case of space curvature: let there be two ships at the Equator, each facing due North. The captains of each ship use a laser ranging device to measure that the ships are 1 mile apart from each other, in the East West direction. Then each one sails due North from his starting location until they reach the Arctic Circle. They take their measurement again, and find that they are significantly closer than 1 mile, as if the ships had some "attraction" to one another. Neither one ever drifted East or West, and there were no forces acting on them. Their paths simply converged, because Earth's surface is curved, not flat.
Now, an example of time curvature. Say I stood on a high tower, say the apex of the KRDK mast, and aimed a laser directly at the ground below me, where you stand with a receiver. I fire a pulse, and then another pulse exactly 1 second later by my clock. You would receive those pulses, and the time between them by your clock would be less than 1 second. The paths of the light rays converged, because the spacetime around the Earth is curved. The effect would be about on the order of $10^{-13}$ seconds at that height, but a very real effect. This has been experimentally confirmed many times.
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$\begingroup$ You say that nearly all the curvature is in time, not space. Does this imply that time dilation gradients are responsible for gravitation due to symmetries in 4D energy vectors? $\endgroup$– elfeiinCommented Mar 3 at 8:15
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$\begingroup$ I don't know what you mean about 4D energy vectors. But yes, there is a sense in which time dilation is gravitation. $\endgroup$– RC_23Commented Mar 3 at 15:22
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$\begingroup$ Sorry, was getting ahead of myself. Is that because particles are more likely to appear where time is slower? $\endgroup$– elfeiinCommented Mar 3 at 17:51
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$\begingroup$ Not at all. It's because the trajectory of an object will be turned towards the direction of slowest time. Look at the squirrel trajectory in the video linked. $\endgroup$– RC_23Commented Mar 3 at 19:59