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I am quite new this relativity, I read some book about this field, and I want to check if I am understanding this topic by considering the following example.

Assume I am staying in a train which is moving ahead with velocity u (u=3/5 c) relative to somebody staying on the platform. I call my frame S' and the frame attached on the platform S. Now I am sitting in the middle of the train and switch the light on. I measure the time when the light hits the front and back end of the train, we call this time t' (same for light hit the front and back end). Of course, the length of the train I measure is L' .

Now someone standing in the platform also measure the length of the train and times for light travel to the front and back end of the train in his frame. He will see the length of the train will be contracted due to

[tex]

L = L' \sqrt{1-u^2/c^2}

[/tex]

And for measuring the time when light hit the front and rear end, in his frame, the distance for the light moving ahead is half of L, and the distance for the light travelling backward (to the rear end) is also half of L. But for the moving-ahead light, the speed is (c-u) and for the backward light, the speed is c (c+u>c, impossible, just use c). So we have

[tex]

t_f = \frac{L/2}{c-u}, \qquad t_b = \frac{L/2}{c}

[/tex]

I don't quite sure this is correct or not. But if I start from the Lorentz transformation, I found

[tex]

t_f = t'\sqrt{1-\dfrac{u^2}{c^2}} + \dfrac{|u|x}{c^2}

[/tex]

[tex]

t_b = t'\sqrt{1-\dfrac{u^2}{c^2}} - \dfrac{|u|x}{c^2}

[/tex]

where t' is the time measure in S' frame, x is half of L.

Obviously, these two methods give different answer. But |t_f-t_b| is the same from these two method. I wonder which one is correct? or both are incorrect?

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# Trying to understand relativity

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