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Ask a Physicist: Time Warp Brain Teaser

Bill, from the US, wants to know:
Would a satellite with a perfectly circular orbit around the center of a circle experience time dilation relative to an observer at the center of the circle? What if the observer were spinning to always be looking at the satellite making them both seem at relative rest? If the satellite does experience time dilation, is it due to the non-inertial acceleration due to centripetal force? 
This is a really insightful question—it applies the concepts of relativity in a very tricky way to create an apparent paradox which might not be obvious at first glance. For those less well-versed in relativity, we'll do a quick breakdown of what this question is getting at.
Relativistic time dilation is at the heart of the question—for an observer traveling at high speeds, time seems to pass more slowly than it does for an observer at rest. The effect is insignificant under Earthly conditions, but produces measurable effects for things like satellites in orbit around the earth, which have to have a very high tangential velocity in order to avoid being sucked in by the planet's gravity.

The picture gets a little more complicated, however, when you consider relativity's most basic incarnation—the standard relativity of motion. If I'm moving away from Earth at 50% the speed of light, I'll age slower than my twin on Earth would. However, if my twin were in traveling the same direction at the same speed, my velocity relative to him would be zero, and we would necessarily age at the same rate—although we'd both be aging slower than everyone on Earth.

So now you see why the question is tricky—if I'm rotating, but the only object in my field of view is orbiting me at exactly my rotation rate, it should look just like the satellite is a constant distance away, at rest. From a non-rotating observer's perspective, the satellite should experience time dilation, since it's moving relative to the stationary center, but from the satellite's perspective (or the central observer's), there's no way to tell that it's in motion...or is there?

Imagine you're standing on Earth at the equator, and directly above your head, there's a satellite in geosynchronous orbit—completing exactly one orbit per day, meaning it stays above your head at all times (as long as you don't move).

So how do you know whether it's at rest or moving? Simple—it doesn't fall on your head!

If your central observer or the planet they're standing on has enough mass to pull the satellite into the circular orbit described in the question, the gravity of that planet would be expected to pull the two bodies together if they were truly at rest with respect to one another. If they were, they wouldn't remain that way for very long, instead accelerating toward each other. 

So, if a massive observer sees an object that appears to be impervious to its gravitational pull, there are only two possible conclusions: either you're looking at a non-gravitationally-interacting satellite (which shouldn't be possible, as even photons are affected by gravity) OR this thing is orbiting you at precisely your rotation rate, meaning it should experience gravitational time dilation.

Thanks for asking, Bill, and keep those questions coming!


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