GPS clocks and special relativity effects

Question

I've got question about global navigation systems. In internet sources i found that atomic clocks in satellites will slow down for 7us per day because of special relativity effects and speed up for about 40us per day because of general relativity effects. I'm interested in special relativity part. For example, if satellite would make circles around some point with same speed as on Earth orbit but without Earth mass in center then there should be only special relativity effects. Will it be in this case 7us slow down? Because it seems strange for me, something like twins paradox. For example, i'm sitting on maseless point and satellite make rounds around me at several kilometers per second. In such scenario satellite clocks should slow down 7us per day. But for satellite my clocks are also slow down because for satellite frame reference i make rounds around him. So then at some point satellite decides to stop make circles and comes to center to my clocks. Does my and satellite clocks measurements will differ? Sorry if similar questions already asked, i haven't found.

Answer 1

Yes, the clocks will differ and indeed the clock moving in a circle at speed $v$ will run slowly by the same factor that it would slow down when moving in a straight line i.e.

$$ \frac{dt}{d\tau} = \frac{1}{\sqrt{1 - v^2/c^2}} $$

I do exactly this calculation in Can a ultracentrifuge be used to test general relativity?

The reason for this is somewhat involved as it requires a deeper understanding of special relativity than is given in introductory courses. I go into this in detail in What is time dilation really? though you may find that goes into too much detail. The bottom line is that if observers move between two points in spacetime then the observer who moves in a straight line will find their clock runs fastest i.e. they measure the most elapsed time on their clock. In this case the satellite and the stationary observer meet up once per orbit, and since the satellite is not moving in a straight line its clock records less time.

We then need to consider what a straight line is in relativity, but actually this has a simple definition. If you are moving then even if you are blindfolded and cannot see your surroundings you can still tell if you are accelerating because you can feel the g forces. The acceleration you can feel is an important property in relativity and is called your proper acceleration. The definition of linear motion is simply that your proper acceleration is zero. Hence roughly speaking the observer with the smallest proper acceleration measures the most elapsed time on their clock.

You may have heard that the resolution of the twin paradox is due to one twin accelerating, and this somewhat vague statement is referring to the proper acceleration. In this case the satellite is experiencing centripetal acceleration due to its circular motion.

Answer 2

The observer in the satellite would not be in a single inertial frame of reference. They are being (continuously) accelerated in a circular motion. Therefore the symmetry between the two observers is broken. If the observer executing circular motion stops and returns to the other observer, they will have aged less when their clocks are compared.

The situation is quite similar to the famous "twins [non-] paradox", where the symmetry between the two is is broken because one must accelerate/decelerate.

Answer 3

But for satellite my clocks are also slow down because for satellite frame reference i make rounds around him.

For satellite it is so that:

1. Your clock is time-dilated
2. The current reading of your clock changes rapidly

There is a simple experiment the satellite, equipped with AI, can do:

Turn off rocket motor for a second, and note that for that time your clock is just normally time-dilated.

Anyway, your clock will be ahead of satellite's clock.