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My text reads thus (the part which is bolded is that which I don't understand, I just reckon that the rest is needed for the context) :

This seems very wrong to me. Am I incorrect in saying that Mavis must see the two flashes simultaneously, and also must conclude that they happened at the same time? She can't take into account her movement towards one of the flashes, because that only happens in Stanley's reference frame. She's at rest with respect to the two sources in her own frame, surely?

Imagine a train moving with a speed comparable to c, with uniform velocity. Two lightning bolts strike a passenger car, one near each end. Each one leaves a mark on the car and one on the ground at the instant the bolt hits. The points on the ground are labeled A and B while the ones on the car are labeled A' and B'. Stanley is stationary on the ground at O, midway between A and B. Mavis is moving with the train at O' in the middle of the car. Both see light flashes emitted from the points where the lightening strikes.

Suppose the two wave fronts from the lightning strikes reach Stanley at O simultaneously. He knows that he is the same distance from A and B, so he concludes that the two bolts struck A and B at the same time. Mavis argees that the two fronts reached Stanley at the same time, but she disagrees that the flashes were emitted simultaneously.

Stanley and Mavis agree that the two fronts do not reach Mavis at the same time. Mavis at O' is moving to the right with the train, so she runs into the wave front from B'beforethe wave front from A' catches up with her. However, because she is in the middle of the passeger car equidistant from A' and B;, her observation is that both wave fronts took the same time to reach her because both moved the same distance at the same speed c. Thus she concludes that the lightning bolt at B' struck before the one at A'

This seems very wrong to me. Am I incorrect in saying that Mavis must see the two flashes simultaneously, and also must conclude that they happened at the same time? She can't take into account her movement towards one of the flashes, because that only happens in Stanley's reference frame. She's at rest with respect to the two sources in her own frame, surely?

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