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 Quote by Michael C Adel, I'm going to stick to the train frame and give some names to things so we can be clear about what we are discussing: IN THE TRAIN FRAME A and B are the two ends of the train. C is the point somewhere between them where there is a light source. p is the event "a light signal is emitted at C". q is the event "the light signal from C reaches A". r is the event "the light signal from C reaches B". What you seem to be claiming is that it could be possible to send a light signal from A after event q that will reach B before event r. Am I correct?
 Quote by Adel Makram yes correct
Without using any relativistic calculations it's easy enough to see that this is impossible. Just put a mirror at A.

But you seem to be claiming that relativistic calculations show this to be possible. In order to understand your argument I need more explanation. In the PDF you just attached, what exactly do you mean by ta and AB ? In what frame of reference are you doing your calculations?