17 The Twin Paradox
181
In order to have something specific in front of us for our following considerations
of the twin paradox, and in order to keep the equations as simple as possible, we will
now proceed to analyse the following case from the viewpoints of all three observers.
These would be the observer B o in o with the coordinates (x, t), twin A in
with
the coordinates (x
, t
) and finally brother A o , who starts in the reference system
o and then moves to the reference system
, where the coordinates (x
, t
) are
ascertained. The hand settings of brother A o ’s personal clock U
A
o will be especially
marked using a tilde, ˜
t. The personal clocks U
A and U
A
o of both brothers have the same
setting, zero, at their common point of departure, the common origin of coordinates,
our event O. They both agree on instigating a neutral observer, B o , who observes
the complete process from the reference system o . Twin A finds himself in the
reference system
and thus moves with the velocity v with respect to the reference
system o , where his brother A o is momentarily positioned at x = 0. Event T then
occurs, as illustrated in Fig. 17.3. Brother A o changes his reference system to
,
which moves with the velocity u with respect to o . Furthermore,
possesses the
velocity u
with respect to
. This situation is illustrated in Fig. 17.5. We now wish to
conditionally determine the velocity u by stating that twin A in his reference system
finds out that: ‘Brother A o is approaching me with the velocity u
= v’. Now,
applying the composition (194), we see for the velocity u =
v+v
1+v·v/c 2
o
the expression
u =
2v c
2
o
c 2
o + v 2 .
(196)
Notice that v < u.
The following expressions of this velocity u will be needed for later use,
1 −
u
2
c 2
o
=
c
2
o − v
2
c 2
o + v 2 ,
1 −
1 −
u
2
c 2
o
=
2v
2
c 2
o + v 2 ,
u − v
=
c
2
o − v
2
c 2
o + v 2 v .
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(197)
We now have the situation as described by Eqs. (189)–(191) for an arbitrary u with
0 < v < u < c o in front of us. We now choose the velocity u =
2v c
2
o
c 2
o +v 2 for the reference system
according to (196) and define, in this situation, the event of reunion
as S(v) and the time measured in o that brother A o spent in
as t v . By applying
Eq. (197), we find for both of the times measured in o ,
t T =
x P
v
,
t v =
x P
v
c
2
o + v
2
c 2
o − v 2 .
⎫
⎪ ⎬
⎪ ⎭
(198)
181
In order to have something specific in front of us for our following considerations
of the twin paradox, and in order to keep the equations as simple as possible, we will
now proceed to analyse the following case from the viewpoints of all three observers.
These would be the observer B o in o with the coordinates (x, t), twin A in
with
the coordinates (x
, t
) and finally brother A o , who starts in the reference system
o and then moves to the reference system
, where the coordinates (x
, t
) are
ascertained. The hand settings of brother A o ’s personal clock U
A
o will be especially
marked using a tilde, ˜
t. The personal clocks U
A and U
A
o of both brothers have the same
setting, zero, at their common point of departure, the common origin of coordinates,
our event O. They both agree on instigating a neutral observer, B o , who observes
the complete process from the reference system o . Twin A finds himself in the
reference system
and thus moves with the velocity v with respect to the reference
system o , where his brother A o is momentarily positioned at x = 0. Event T then
occurs, as illustrated in Fig. 17.3. Brother A o changes his reference system to
,
which moves with the velocity u with respect to o . Furthermore,
possesses the
velocity u
with respect to
. This situation is illustrated in Fig. 17.5. We now wish to
conditionally determine the velocity u by stating that twin A in his reference system
finds out that: ‘Brother A o is approaching me with the velocity u
= v’. Now,
applying the composition (194), we see for the velocity u =
v+v
1+v·v/c 2
o
the expression
u =
2v c
2
o
c 2
o + v 2 .
(196)
Notice that v < u.
The following expressions of this velocity u will be needed for later use,
1 −
u
2
c 2
o
=
c
2
o − v
2
c 2
o + v 2 ,
1 −
1 −
u
2
c 2
o
=
2v
2
c 2
o + v 2 ,
u − v
=
c
2
o − v
2
c 2
o + v 2 v .
⎫
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭
(197)
We now have the situation as described by Eqs. (189)–(191) for an arbitrary u with
0 < v < u < c o in front of us. We now choose the velocity u =
2v c
2
o
c 2
o +v 2 for the reference system
according to (196) and define, in this situation, the event of reunion
as S(v) and the time measured in o that brother A o spent in
as t v . By applying
Eq. (197), we find for both of the times measured in o ,
t T =
x P
v
,
t v =
x P
v
c
2
o + v
2
c 2
o − v 2 .
⎫
⎪ ⎬
⎪ ⎭
(198)