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How does a clock go slower if it is travelling at high speed?

Einstein says that a clock travelling at high speed (near the speed of light) will go slower than a `stationary' clock. It has been demonstrated that a clock returning to earth from an orbiting satellite shows less time than a stationary, earth bound, clock. How come? Velocity is relative. I.e. the earth bound clock is moving at high speed w.r.t. the satellite bound clock, such that the latter could be considered stationary, and the earth bound clock to be moving at high speed. So why doesn't the earth bound clock go slower?

Your thinking of the twin paradox in special relativity. The problem with the twin paradox, is that it doesn't work with special relativity alone, you need general relativity to make it work properly.

In the twin paradox, one twin stays on earth and the other flys to a distant star at a high speed, turns around, and comes back to earth. The twin on earth sees the twin traveling at a high speed as having a slower clock, so when that twin comes back to earth, he's younger than the twin on earth. Special relativity ignores the perspective of the twin traveling in the rocket ship because that twin is not in an inertial frame. When he turns around, there is tremendous acceleration, from close to the speed of light in one direction to close to the speed of light in the other. Special relativity can't deal with acceleration.

In general relativity, you can look at the perspective of the twin in the rocket ship and find that when he turns around he sees the twin's clock on earth as moving very very fast, so when he gets back to earth, they end up agreeing that the twin on the ship is younger.

It's similar for satellites. They accelerate around the earth and experience less gravity, this affects their perspective differently and it cannot be properly accounted for with special relativity. You need general relativity. In general relativity, different reference frames aren't equivalent because some are more or less inertial. Frames that have the same inertia are equivalent. Others are not.

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