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This question already has an answer here:

Concerning gravitational time dilation in artificial gravity (made by a rotating torus like in many sci-fi movies) how would you go about calculating the effect?

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marked as duplicate by ACuriousMind, Kyle Kanos, John Rennie, RedGrittyBrick, Carl Witthoft Jul 31 '15 at 11:45

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  • $\begingroup$ Hi Joe. If you look at my answer to the question I've linked I explain how to calculate the time dilation for rotational motion. It's for a centrifuge rather than a space station, but the calculation is exactly the same. As Ed says in his answer, the time dilation is the same as time dilation for straight line motion at the same speed. $\endgroup$ – John Rennie Jul 31 '15 at 6:20
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Since there is no excess mass, there is no deformation of spacetime and thus no gravitational time dilation. However, since there is a velocity there is a special relativistic time dilation. Particles going around in circular accelerators only experience special relativistic time dilation.

The time dilation from the motion will be equivalent to a gravitational time dilation with effective mass of $M_{eff} = \frac{v^2r^2}{G}$ (where r is the radius of the rotating torus). And there is a major difference between this time dilation and the usual special relativity time dilation with inertial frames. Since anyone or thing in the torus is undergoing constant acceleration it is constantly changing inertial frames. This motion is therefore not relative and the clocks in the non-rotating frame appear to the entities in the rotating frame to be going faster. Thus, when a twin leaves the rotating frame, he/she will actually be younger than the stay at home twin.

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  • $\begingroup$ I always thought that if a person were in a closed room that was being accelerated up they would have no way of detecting if the room was being accelerated or if it was gravity holding them to the ground. Based on your answer I would say you could test for gravitational time dilation and then see that it is not there. Am I right with assuming this? $\endgroup$ – Joe Jul 30 '15 at 22:57
  • $\begingroup$ The rule is that without looking outside the room, you can not tell. And that is so here $\endgroup$ – eshaya Jul 31 '15 at 1:20
  • $\begingroup$ An ascending elevator and a rotating torus are two distinguishably different systems. There are no Coriolis forces in an elevator, for example. $\endgroup$ – Asher Jul 31 '15 at 3:10
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    $\begingroup$ Ed's answer is correct, though it would be nice to see the working behind the answer. As it stands, the answer is not going to improve anyone's knowledge of relativity. $\endgroup$ – John Rennie Jul 31 '15 at 6:21
  • $\begingroup$ @Asher It so happens that I asked John Wheeler about this very issue of detecting coriolis forces and/or tidal fields to distinguish the origin of the acceleration (in a 1976 GR class). His answer was that when Einstein spoke of doing experiments within a closed room, he meant a very small room, essentially at a point in space. The distinguishing experiments require two points separated by some distance. $\endgroup$ – eshaya Jul 31 '15 at 16:06

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