Time dilation contradiction Let $S$ and $S'$ be two frames, moving with velocity $V$ relative to each other. Treat $S$ as a stationary frame, then $S'$ moves away with velocity $V$. Let $\gamma = 7$ so that $\Delta t = 7\Delta t'$. Does this mean that if there are exact clocks in each frame and observers in $S$ establish a video conference call with observers in $S'$, then the clock in $S'$ will show time moving seven times slower?
I don't think that's going to be the case because we can treat things the other way around by taking $S'$ as stationary.
But then, what about time dilation? How should it happen in this case? Will clocks in $S$ and $S'$ differ at all?
 A: 1)  Suppose Alice sits on earth while Bob travels away at speed $v$.  
2)  First do everything in Alice's coordinates:  Bob is traveling along the line $x=vt$.  Alice, at time $t_0$, sends him a light signal that travels along the line $x=t-t_0$.  To see when it arrives, solve $x=vt=t-t_0$ to get 
$$t={t_0\over 1-v}\qquad  x={vt_0\over 1-v}$$ 
3) Now Lorentz-transform to Bob's coordinates.  This gives
$$t'={t-xv\over\sqrt{1-v^2}}={t_0(1+v)\over\sqrt{1-v}^2}$$
4)  Therefore the speed of the video stream arriving at Bob's ship is slowed down by a factor of $(1+v)/\sqrt{1-v^2}$.  This factor is not just your $\gamma$, because that fails to account for the fact that Bob is moving away from Alice as the video streams.
5)  By symmetry, exactly the same is true of the speed of the video stream arriving at Alice's screen from Bob's ship.
6)  As to your (separate) question about whether Bob's and Alice's clocks differ, that depends, of course, on who you ask.  Alice says Bob's runs slower than hers, Bob says Alice's runs slower than his, and Ted, who is moving away from Alice in the same direction as Bob at speed $v/2$, says they don't differ at all.
