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So imagine two lightning bolts hit the ground, simultaneously to a stationary observer. There is also a person on a train traveling to the right at a constant velocity. I know that if he started in the midpoint, he would see the light from the right first because he is shortening the distance. I know the equation for calculating the time difference is $t'_2-t'_1= \gamma[-\frac{V}{c^2}(x_2-x_1)]$.

However, I am very confused because the time difference only depends on the positions of the lightning strikes and the velocity of the train. But if the train was to the left of the leftmost lightning strike, the left light would reach him first which is the opposite of what happens when he is between them. I am confused because this doesn't change the equation at all. Obviously changing x' should change this equation, but I don't know why or how.

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The position of the train will affect the time difference between when the person receives the light from each of the lightning strikes. However, this is different from the time difference between the strikes in the train's frame. This is because the person can calculate how much time it took for each light to reach them from the positions of the strikes, and use that to determine the times of the strikes.

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