I Information transfer between moving bodies - sped up or slowed down?

[some bloke]

TL;DR

If something orbits earth at a fast enough speed for an appreciable relativistic effect, would information it receives from earth (like radio communication) be shifted to be faster or slower?

Relativistic motion and time dilation has been of interest to me for a long time, and whilst I am asking this for realism in writing a piece of fiction, I am curious as to how this interaction would work.

Let's say there's a large dyson ring built around earth, which is spun up such that the ring approaches the speed of light relative to the surface of earth, and so experiences time dilation. I am familiar (though confused) with the concept that two objects moving past each other would each perceive the other to be experiencing time slower than themselves (I may have garbled this and will welcome correction!) but the main effect I'm using here is the "people getting off the ISS are 0.01s younger than if they stayed", but ramped up. I am thinking "people who stay on the ring for 5 years only age 1 year" sort of thing. For now, ignore all the centipetal forces which would rip them to sludge!

My question is this: If the ring were picking up radio signals from earth, would they be received at 5x the speed (EG a live broadcast which lasts 5 minutes, would it take 1 minute on the ring?). Similarly, if the ring sent radio signals to earth, would they be received at 1/5 the speed they were sent?

My goal is to accurately(ish) explain remotely controlling something on earth from the ring, so needing information to pass in both directions. I assume that this will be limited by the slowest direction, EG if you are typing on a computer remotely from the ring, you would have a 5x lag on anything you do?

Anything you can offer to help me understand it gratefully received!

 

[Ibix]

Time dilation is symmetric between inertial (i.e. non-accelerating) frames. But the rest frame of your ring is non-inertial, so don't expect that to hold.

There's no general prescription for dealing with non-inertial frames, but this one turns out to be easy. Clocks on the ring tick slowly compared to clocks on the Earth from both the Earth frame and the rest frame of the ring. There's a complexity with gravitational time dilation for realistic orbital velocities (gravitational time dilation makes the higher clock tick faster, the orbital speed makes it tick slower, and which effect is larger depends on altitude), but here your rotational speed is so high the gravitational time dilation is negligible.

What happens to radio frequencies is messier to compute. You get changing frequencies each orbit as the ground station and orbiting stations' velocities change with respect to each other. The details depend on altitude and speed of the ring and the latitude and longitude of the ground station.

 

[Nugatory]

Anything you can offer to help me understand it gratefully received!

Even before you consider the relativistic (which @Ibix has summarized for you, and do not underestimate the amount of thought that precedes "...this one turns out to be easy") you should think through the Doppler effect.

Imagine that on both earth and the ring there are strobe lights, identically constructed to flash once a second as measured by someone sitting right next to them. Now, even without considering any relativistic effects, you can work out that when the two are approaching one another they both see the other strobe light flashing fast relative to their own (each successive flash has to cover a shorter distance between emission and reception)b; and when they are moving away from each other they both see the other strobe flashing slow relative to their own. Which we get at any moment depends on where in the orbit we are.

Now imagine that these instead of strobe light flashes these are the crests and troughs of radio waves coming by, or because you are interested in the information transfer rate wave packets each carrying a bit of information.