I didn't actually needed the math, but its beautiful

@G.Smith I provided such details... let me know if you need more

Does that mean that in the perspective of a photon, any travel is instantaneous?

thanks for the explanation! can i abuse it a little more? having $v$=($c$-1) and T=1(in years) we have B's clock showing a really small number. I must have messed up my math or my understanding of the whole concept, because such result imply that B experienced a very small trip, but we do know that he traveled the distance of 1 light year! Where am I wrong? Math or understanding?

wow! such a nice answer! can you pls just fill in the gap for me?: given the scenario where B was already moving at $v$ at time zero what would the clocks actually show when compared?

@ToddWilcox exactly in the middle of the way between A and B theres a light bulb. When A and B detect that the light bulb turned on, the experiment begins... happy? honestly im proposing a engine that accelerate almost $c$ instantaneously...

@ToddWilcox sorry Todd but I seem to miss your point... as i understand, stopwatch is the kind of watch that start from zero and count forward... as in en.wikipedia.org/wiki/Stopwatch... and they will be started at the same time because both observers agreed that upon a event that will happen at the same time for both will be the start of the experiment...

@DanielSank sorry about that! out of nowhere 20+ edits came along! I must have missed yours... thanks for the corrections...