Natural Science Forum / Physics / Relativity / March 2005

There might be hundreds of questions about time dilatation like this,
but I couldn't find a similar problem in the groups.

Suppose a rocket A travels with v=4/5c for 5 years (y) from earth to a
planet (in rest relatively to earth) which is 4 light years (cy) away
and then returns to earth with v=4/5c, then the time on earth elapsed
is t_E=5y+5y=10y. With gamma1=3/5 the time elapsed for the rocket crew
is
t_A=3+3=6y.
We can not see the things the other way round, since the rocket is no
inertial system. OK?

Now, a rocket B travels together with A (same direction & speed) to the
planet but then carries on with the same speed. Then B is an inertial
system, so we should get the same ratio of elapsed times, shouldn't we?

What B should see:

1.
Travelling to the planet with A takes 3y, the distance is, because of
Lorentz contraction and gamma=3/5, 12/5 cy. The time elapsed on earth
is t_E1=3y*gamma1=9/5y=1.8y, i.e. for B is
t_A1=3y, d_A1=12/5cy

2.
When A travels back to earth, the velocity relatively to B is
u=2*v/(1+v*v/c^2). v=4/5c and therefore u=40/41c and gamma2=9/41.
With this speed it needs for the distance planet-earth of 12/5cy (which
is still the same)

t_A2(B)=(12/5cy)/(40/41c)=123/50y=2.46y (for time in system B) and
t_A2=t_A2(B)*gamma2=27/50y=0.54y (for time in system A)
t_E2=123/50y*gamma1=369/250y=1.476y.

t_A=t_A1+t_A2=3.54y, so the rocket crew A aged 3.54 years for B.

The total elapsed time on earth is t_E=t_E1+t_E2=819/250y=3.276y years
for B.

I was expecting the same ratio t_A/t_E. Is it correct that, for B, the
return is faster than the first part? I know, I probably did more than
one mistake, but maybe we can sort is out.

Any help is being appreciated.

Ralf