Time dilation for satellites

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?

• The satellites are not attempting to navigate. So what we see matters, and what the satellite sees does not. There's your asymmetry. – WillO Apr 16 '17 at 22:47

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.

"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