In Special Relativity relativity of simultaneity is the fact that 2 simultaneous events occurring in a stationary frame does not appear simultaneous in a moving frame. For example, in Einstein’s train thought experiment 2 simultaneous flashes of light on the platform do not appear simultaneous for the observer in the train. But relativity of simultaneity has never been tested with real simultaneous events.
For testing relativity of simultaneity we need 2 synchronized clocks moving at high speed and we will read them in a stationary frame. Fortunately, we have at hand many GPS satellites which carry precision clocks and broadcast their time, with which we can check relativity of simultaneity.
Suppose that 2 satellites are separated by the distance L in the same orbit. Their clocks are synchronized with one clock on Earth, that is, the event “time of the satellite 1 is t0” and the event “time of the satellite 2 is t0” occur simultaneously on Earth at the time te. In the frame of these 2 satellites, due to relativity of simultaneity, these same events occur at time t1 on the satellite 1 and t2 on the satellite 2 and the difference of time is dt= t2- t1.
Suppose that we have n satellites equally spaced in the same orbit which is circular. The nth satellite is the last satellite and the (n+1)th satellite is the first satellite, which complete the circle of the orbit.
Due to relativity of simultaneity, the difference of time is always dt from one satellite to the next and the difference of time between the ith satellite and the first satellite equals (i-1)*dt. Then, the difference of time between the (n+1)th satellite and the first satellite equals n*dt. So, The time of the (n+1)th satellite is t1+n*dt.
We notice that the (n+1)th satellite is the first satellite but its time, t1+n*dt, is different from t1 the time of the first satellite. How can the time of a satellite is not the time of itself?
I explain this phenomenon in the article below.