Two objects are moving towards a gravity well. They are at $x$ distance from each other and moving at a fixed speed. The gravity well is massive enough for the objects to experience significant time dilation when near the well. As the first object approaches the gravity well, it will appear slower to the second object, which is experiencing less severe time dilation. Does this mean that the second object will be able to catch up the first object? Will the two objects arrive closer to each other than when they began traveling?
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$\begingroup$ There are a number of issues with your problem statement -- for one thing both "distance" and "speed" are relative concepts, and for another the speed cannot be fixed if the objects are in free fall. $\endgroup$– Eric SmithCommented Apr 30, 2023 at 14:22
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$\begingroup$ Moving at "fixed speed". What does that mean? Fixed speed relative to what? How measured? How constrained? $\endgroup$– John DotyCommented May 1, 2023 at 13:47
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1 Answer
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A ruler free falls downwards. Do observers observe the length of the ruler changing? Yes, all those observers that observe the speed of the ruler changing, observe the length of the ruler changing, accordingly to the Lorentz-contraction formula.
Now we may say this phenomenon is the Lorentz-contraction. Or we may say it's the stickynotememo-contracion, as it's derived using general relativity, while the Lorentz-contraction is derived using just some special relativity.
So a ruler placed between two free falling objects says that the objects stay at constant distance from each other. Said ruler was free falling with the objects.
Oh yes, here "speed" means how fast an object moves compared to how fast light moves at the same place.
