Aer said:
Everything I said is completely correct regarding special relativity.
'Fraid not.
Special relativity says that the Earth clock ticks
faster than the satellite clock in
all frames.
Not if my satellite zooms around the Earth at 0.5c for instance and my frame is the satellite rest frame.
Careful! In special relativity, the satellite rest frame is not inertial, and a naive application of the time dilation formulas is not appropriate.
Consider the following scenario, and an analysis using simultaneity:
Consider two clocks, A and B.
The rest frame of A is inertial.
B is moving in a circular path with A at the centre, with a constant speed of 0.5c relative to A.
Let's consider simultaneity in the rest frames of A and B.
Since the direction of A from B is always perpendicular to A's velocity relative to B, and the direction of B from A is always perpendicular to B's velocity relative to A, we see that according to special relativity:
If an event at A is simultaneous with an event at B in A's frame, then the two events will also be simultaneous in B's frame.
This immeidately implies that time dilation between A and B must be absolute. The two clocks
must agree on which one is slow:
Say A's clock resets to zero simultaneously with B's clock resetting to zero in A's frame. We know that these reset events are also simultaneous in B's frame.
Now say that A's clock reads 100s simultaneously with B's clock reading 86.6s. We know that these ticks are also simultaneous in B's frame.
So in A's frame, 100 seconds pass in the time it takes 86.6 seconds to tick past on A's clock.
And in B's frame, 86.6 seconds pass in the time it takes 100 seconds to tick past on A's clock.
There ya go.