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We have one frame of reference - a train moving from left to right with constant speed v relatively to the embankment, and second frame of reference - the embankment itself. On the embankment there are points A and B and their midpoint M. On the train there is the point M'. When M and M' meet each other, two lightnings strike both A and B. The observer on the embankment sees that the two flashes of light meet at the midpoint M. But since the train is moving and the point M' with it, M' moves towards B and therefore the observer on the train will see that the beam from B will arrive first at point M' and after that will arrive the beam from A. And so simultaneity is relative - for one observer the two events are simultaneous, but for the other they are not.

But lets imagine for a moment that when the points M and M' coincide we put points A' and B' on the train which coincide respectively with A and B. So it is the same whether we say the beams start at A and B or at A' and B' - at the moment of the lightning strikes, those points coincide with each other.

Lets imagine the people on the embankment and those on the train are aware of the experiment that is taking place. What is the point of view for the people on the embankment? They know that the speed of light is constant c in every direction, and therefore when the point M and M' meet they begin to wait for the two flashes of light and expect the light beams to meed at their midpoint M - the light needs equal time to travel the equal distances AM and BM. So they are right for themselves.

Now lets consider the point of view of the train passengers. They know about the principle of relativity and so too expect the speed of light to be constant c in every direction. At the moment when point M and M' meet, the people start to wait for the flashes too. They know that point A' and B' are equally distant from the point M', because when M and M' coincide(and the flashes occur) - A and A', and B and B' coincide too(whether we take length contraction into account or not shouldn't matter because the important thing here is that the length A'M' is equal to B'M' according to the train passengers - according to them A and B are equally distant from M, and M' and M coincide at that moment with or without length contraction). So knowing that A'M'=B'M' and expecting the speed of light to be the same both from A' to B' and from B' to A', they would expect the light to cover the distances A'M' and B'M' for the same time, and therefore meet at point M'.

What prevents the passengers from making such conclusions, and not be surprised when the two flashes don't meet at their midpoint M'? What is the difference between the train and the embankment - surely we can say that the train is stationary and the embankment is moving relatively to it with velocity v, so when the flashes occur at points A' and B', the point M will be moving towards A' and the people on the train will expect that for the people on the embankment the flashes won't be simultaneous. So is this difference in the way the two beams arrive at points M and M' real - for the people on the train the beams will meet at some other point? Or it is only a matter of relative conclusion - the observer on the embankment will expect the beams will meet only on his midpoint, and the observer on the train will expect the same thing?