binis said:
Τhe principle of relativity, is that there is no such thing as "earth motion" or "satellite motion", only relative motion.The satellite movement is equivalent of the earth's.Therefore ds=ds' Satellite is almost immobile relative to the center of the Earth,differently to a point on the surface.
As many posters have already stated, this only applies when dealing with inertial frames. Once you start to make measurements from a non-inertial frame, this changes. If you are in the tail of an accelerating rocket, for example, you will measure a clock in the nose of the rocket ticking faster than your own, even though you have no relative motion between you and it. (conversely, someone in the nose will measure your clock as running slow). And this is not confined to clocks in the rocket or sharing your acceleration. You will measure any clock "ahead" of the accelerating* rocket as ticking fast, the further away, the faster it ticks.
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Now take this rocket and arrange things so that its circling a clock at a constant speed, ( it is pointing its rockets outward in order to maintain a "forced orbit" around the clock). It is still accelerating, even though its speed remains constant. The acceleration is centripetal and always pointed at the central clock. As such, someone in the rocket would measure the central clock as ticking fast.
Someone at the central clock would be in an inertial frame and it would only be the rocket's circular velocity that would effect how they would measure the rocket's clock tick, and this would cause them to measure it as ticking slow. The point is that, while both the Central observer and rocket observer can lay claim to being "at rest", only the central observer can do so from an inertial frame, the rocket observer's "rest frame" would be a non-inertial rotating one, for which there are additional factors beyond just relative motion for determining relative clock rates.**
With GPS satellites, gravity provides the centripetal acceleration, and the difference in gravitational potential adds an additional complication, but the idea that the only important factor is "relative velocity alone" still does not hold.
* in this case "ahead" means in the direction of the acceleration
** You could also have the central clock rotating so that it maintains the same orientation with respect to the rocket at all times, and consider itself as being at rest within the same rotating frame as the rocket, But now it would be like the Nose clock in the first accelerating rocket example, and would still measure the rocket clock as ticking slow.