The Lack of Simultaneity between moving observers
When two events occur at the same time, they are said to have occurred simultaneously. However, what appears to occur simultaneous to one observer will not appear simultaneous to another observer traveling at a high velocity relative to the first observer. This is because each observer will see lengths contracted and time slow down in the other observer's system. To examine the lack of simultaneity between two systems moving with a high relative velocity between them, we will use a train passing a platform. Observer B is on the center of the moving train and an observer A on the center of a station platform. I have added clocks to further illustrate the lack of simultaneity. When the train is standing still it is 300 meters long. In fig. 1 we see the train standing still at the station platform. Observer B will always measure his train as 300 meters long, whether there is no movement or a constant movement between him and observer A. It takes light one micro-second (μsec) to travel from one end of the train to the other. One micro-second or 300m/c is our time unit, TU. All the clocks show time in TU's (micro-seconds). The platform is 240 meters long.
How observer A to sees the lightning flashes
In fig, 2 the train with observer B is moving past the station platform and observer A with a constant velocity of 0.6c. Observer A is in the center of the station platform. The platform is 240m long, with a clock at each end. Because the train is moving past A with a velocity of 0.6c, to A the train length L will be contracted to 0.8 of its rest length (Lo) 0.8 x 300m = 240m, The same length as the station platform. The time on the train's back clock is offset ahead of the front clock by v/c2 times the clock's rest distance between the clocks. Here c = 300m/μsec and the clock offset is Lo(v/c2) = Lo(v/c)/300m/μsec. Hence the clock at the back of the train will be 300m(0.6)/300m/μsec = 0.6μsec ahead of the front clock.
Observer A will see the lightning strike both ends of the train simultaneously at 0.4 TU (t1) after the event occurred. He sees the readings all the clocks that were lit up at that instant of both lightning strikes. Both the clocks on the platform read 1.0. However, he sees the clock at the front of the train reads 1.0 but the clock at the rear of the train reads 1.6 TU.
What observer B sees
Observer B sees his train length as 300m and his clocks in sync at the ends of the train (150m either side of him). He sees the platform moving past him with a velocity of 0.6c. Observer B should see the platform and everything on it as contracted to 0.8 the size that observer A sees them. He sees the platform clocks as being 0.8x240m = 192m apart. That is observer B sees the platform as much shorter that his train.
Observer B sees the clock at the front of the train is aligned with clock on the right side of the platform when the first lightning strike occurs. This is shown in fig. 3. Then 0.6 TU later, after the platform has traveled 108m, (0.6x0.6x300), B sees that the train's back clock has now aligned with the clock on the left side of the platform and the second lightning flash occurs. This is shown in fig. 4. Note that 192m + 108m = 300m is the length of the train.
The Minkowski diagram illustrates lack of simultaneity
The representation of simultaneity on the x,t plane of the Minkowski diagram is simple. The events will be simultaneous if they have the same time coordinate in the given inertial system. In fig.5 the events E1 and E2 occur simultaneous in the observer's system but not in the object's system. In fig. 6 the events E'1 and E'2 occur simultaneous in the object's system but not in the observer's system.
We have seen that observer A sees the lightning flashes as occurring simultaneously. He sees his own clocks as being synchronized and the clocks on the train as being progressively out of sync. He sees B's length contracted and time dilated. Observer B sees the lightning flashes occurring 0.6 time units apart. He sees his own clocks as being synchronized and the clocks on the platform as being progressively out of sync. He sees A's length contracted and time dilated.