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I read that satellites is effected by the time dilation caused by gravity and also by that one from special relativity. And so there is a need to prepare the onboard clock to ensure that the time is synchronized with a clock on Earth.

But why is this effect not symmetric? The satellites should see that the clock on earth is slowed down and vice versa?

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    $\begingroup$ The satellites are not attempting to navigate. So what we see matters, and what the satellite sees does not. There's your asymmetry. $\endgroup$ – WillO Apr 16 '17 at 22:47
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It is true that satellites are affected by both types of time dilation (that due to motion & that due to gravity), but neither of those types is symmetrical in this situation.

Gravitational time dilation is never symmetrical; observers of each other will always agree on which one has the faster (or slower) rate of time passage.

The time dilation due to motion would be symmetrical if two observers are each in inertial reference frames, but that is not the case here. Observers on Earth can be considered to be in an inertial frame, but the satellite is continually accelerating, so its frame is noninertial.

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"The difference between the classical and relativistic treatments appears more clearly when we consider a receiver moving uniformly in a circle around a transmitter, or vice versa (in flat spacetime). This is a stationary configuration (up to spatial isotropy), so we can definitely say the spatial distance traveled by the signal pulses is not changing. Classically it would follow that there was no Doppler effect, but the time dilation of special relativity implies that the proper time of the circling entity is reduced relative to the rest frame time coordinate of the central entity. As a result, the received signal will be either red-shifted or blue-shifted, depending on whether the transmitter or the receiver is moving in a circle." http://mathpages.com/home/kmath587/kmath587.htm

Rotational motion: https://www.youtube.com/watch?v=hnphFr2Iai4

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A low flying satellite's point of view:

I see landmasses below me, ahead of me, and behind me. All those landmasses are moving, so they are all time-dilated. Landmasses below me are not moving towards me or away from me, so I see those landmasses as time-dilated as they are. Landmasses ahead of me are moving towards me, which causes an effect where I see clocks on those landmasses running faster. Landmasses behind me are moving away from me, which causes an effect where I see clocks on those landmasses running slower. There are slightly more landmasses ahead of me than behind of me, so there would be slightly more clocks that I would see running fast than clocks that I would see running slow, if the land was evenly covered by clocks.

So, on the average the clocks tick a little bit faster than normal, and one clock's average ticking rate during a long time is a little bit faster than normal.

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