Effect of time dilation in Cs atomic clocks? Let's say, Observer A is at rest and Observer B is in a spaceship moving at a significant fraction of speed of light. B synchronized his atomic clock with A's when his spaceship just passed A.

Now, some time later a star becomes a supernova and A and B observe (record the time coordinate) of this event supernova.  The exchange messages to compare the time they recorded with each other and they see that they measured different times for the supernova event.
Cesium's transition frequency (9,192,631,770) does not change, for A and B. So, how did the measurements change?
 A: A sees that the calcium clock of B runs slower, and viceverza. But the problem is not symmetric. First, the supernova is moving relative to one of the observers. That observer will see that the distance betweem him and the supernova is smaller than what the stationary observer will see. Second, at least one of the observers has to accelerate if they are going to meet again. The acceleration changes things in such a way that the cesium clock has marked less ticks for the moving/accelerating observer. To him, the cesium clock of the stationary observer was running slow when he moved at constant speed, and then accelerated when he was decelerating. 
Case with no acceleration: Asumming that the star explodes when the distances to both is the same according to A, then A will see that B marks a larger time than him. Even if to him B's clock is running slower, he see that B marks the explosion after it happens. To A both events, the explosion and B writing the time of the explosion, are not simultaneous.
B will also see A's clock as running slower than him, and will also see that the explosion happens before A marks the time. But to B the explosion happens when the star is closer to A than to him. 
A: 
Cesium's transition frequency (9,192,631,770) does not change, for A and B. So, how did the measurements change?

The measurements didn’t change, they were just measuring different things. If I measure how many meters away London is from me and you measure how many meters away London is from you then even though our meter sticks do not change we will get different measurements. This is because the distance to London depends on the reference point, and our two measurements were using different reference points. 
Similarly, the time of the supernova depends on the reference velocity and the two measurements were using different reference velocities. Nothing changed about the measuring devices, they were just used to measure different things. 
