Billy T said:
The explosions were photo-electrically trigered by a flash of light originating from the center (outside also) of the train, so simultaneous in the train frame, but the ground observers find that the "end of train explosion" preceeds that "head of train explosion" because while the light is traveling to the "end explosion" the end of the train is advancing to meet the advancing flash of light and like wise the front explosion is delayed as the flash must travel the extra distance the train advanced while flash was in motion.
In your example, you indeed assume that in the train both flash light travel at the speed of light. But you also assume that in the ground frame, the velocities of the flashlights are not c-v and c+v, and you do not have reasons to assume it, unless you say that galilean transformation (which lead to the galilean formula of addition of velocity where the flashlights would have velocities c-v and c+v and would have arrived simultaneously to both ends of the train) is incorrect ==> time is relative. So that in fact your proof of non universality of time goes like that: Time is relative (I don't use galilean transformation) so the train experiment shows that with this assumption time is relative.
According to you, since you do not use the galilean transformation, you should have stated what velocity addition formula you use, because in principle you could have stated that the flashligth going toward the front end of the train travels or a ground observer at, let's say, 2c and the flashlight going backward goes at c/2.
Of course, if you say as SRT says that the speed of light is constant in all frames, the example that you gave is perfectly good to show that time and simultaneity are relative.
Futhermore. I totally agree with you that the relativity of simultaneity is a real effect. People who don't understand SRT sometimes think that this relative simultaneity is just an illusion because we "see" the events by mean of light which has a finite velocity. This effect of "seing" is a Dopplerlike effect which would give different result wether we use SRT ore wether we don't.
Finallt, i would like also to point out that I had the impression that people here think that if simultaneity is relative, then every thing is permitted. Well, this is not.
For example, if two events are simultaneous in some frame at the same location, then they will be simultaneous in all reference frames. This means that if particle 1 reaches point A at some time and particle 2 reaches the point A at the same time (this means that they collide), then in all frames they will reach the point A at the same time.
Another example is that if two events have a timelike separation, in all frames the order of the events will be preserved. If a particle reaches point A and later reaches point B, if this particle goes slower than light (the two events have timelike separation), then in all frames the particle will reach point A before point B