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 t`
_{a} and AB` ? In what frame of reference are you doing your calculations?