**Problem : **
A spacecraft travels at 0.99*c* to a star 3.787×10^{13} kilometers away. How long will a
roundtrip to this star take from the point of view of someone on the earth?

If we calculate the number of seconds in a year it turns out that

3.787×10^{16} meters is about 4
light-years (the distance light travels in one year at

*c*). The spacecraft is traveling virtually at

*c*, so the
trip to the star takes 4 years of earth time. The roundtrip takes 8 years.

**Problem : **
With reference to the previous problem, how long will the roundtrip take for someone in the spaceship,
according to someone measuring from the earth?

According to an observer on the earth, since the spacecraft is moving, its passengers' time is dilated. The
factor by which this occurs is

*γ* = = 7.09. The passengers measure

* less * time so, the roundtrip time is

(1/7.09)×8 = 0.14×8 = 1.1 years.

**Problem : **
Now in the reference frame of someone in the
spaceship, what is the time taken for the roundtrip
as observed by a passenger, and by someone on earth (ignoring the times when the spaceship is accelerating
or decelerating).

The whole point of the twin paradox is that a passenger on the spaceship apparently measures the opposite:
that is, that the trip takes 8 years for them, but only 1.1 years for those standing back on the earth. It turns
out that this reasoning is incorrect and in fact the passengers measure the same times as an observer on the
earth when the (General Relativistic) effects of acceleration and deceleration are taken into account.

**Problem : **
If one person stays on earth and one person travels to the distant star, who will age more during the trip and
by what amount?

As we have seen, the reasoning of the passenger on the spaceship is erroneous because the spaceship is not
in an inertial reference frame. The reasoning
of the person on earth is correct (the earth is
approximately inertial). They measure the passenger as aging less than themselves by an amount

8 - 1.1 = 6.9 years.

**Problem : **
Twin A floats freely in outer space. Twin B flies past in a spaceship at speed *v*_{0}. Just as they pass each
other they both start timers at *t* = 0. At the instant of passing B also turns on his engines so as to
decelerate at *g*. This causes B to slow down and eventually to stop and accelerate back towards A so that
when the twins pass each other again B is traveling at speed *v*_{0} again. If they compare their clocks, who
is younger?

This is just a variation of the same problem (that is, the twin paradox as stated in
Section 2). Twin A is in an inertial
reference frame so she can successfully apply the logic of
Special Relativity to find that B's time is dilated and hence that B is younger. B is not in an inertial reference
frame so the opposite reasoning does not apply, and we conclude that when all the effects of the acceleration
are accounted for he must agree with his twin that he is younger.