A rocket passes the earth and synchronises its clock with the earth. Years later, a rocket passes it going to the earth and synchronises its clock with the first rocket. When it reached the earth, will the clocks be synchronised?
Let's say that the Earth is stationary, the two rockets have the same speed v and the refrence planes are: K for Eath, K' for rocket1 and K'' for rocket2.
We can separate the problem in to two parts.
1st part: The rocket goes from earth to a distance $L$(in K). Because the time $t0$ is the same, at distance $L$ the time $t1'$ is equal to $t1'=γ(t1 - vx/c^2)$. Now at this exact same time the second rocket pass and gets time $t1'$ from the other rocket($t1'=t1''$)
We want to find the relation betwen $t2''$ and $t2$.
Now for K: $L=v(t2-t1)$(1) and K'': $L''=-v(t2''-t1'')$(2) This hapens because in K'': $L''=γL$, the objerver in K'' things he is stationary and the earth approches him with speed $-v$
So: $L''=γL$ and $t1'=t1''$
$(1)/(2)==> 1/γ=v(t2-t1)/-v(t2''-t1'') t1''-t2''=γ(t2-t1) (t1-(vL/c^2))γ-t2''=γ(t2-t1)$
Solving the equation with $t1=L/v$, we get: $t2''=γ[(2/v^2 - 1/c^2)L -t2]$