176
17 The Twin Paradox
The later therefore aged less during his journey (which should compensate him
for losing the farm)—stays young and drives fast.
This result is from twin A’s point of view of course self-evident, because it is
brother A o who moves. And it is the moving clock that goes behind. The velocity
with which A o departs from A and the then following arbitrary velocity chosen by
A o to return to twin A should make no difference to the sign of this effect, of course
from out of the view of A in the reference system
. However, what does this case
look like if it is observed from the reference system o , the reference system in
which our observer B o controls the personal clocks of both twin brothers?
We will accept that it is once again twin A who moves with the constant velocity
v with respect to the reference system o . Brother A o sits and watches, together
with observer B o who makes all of his measurements in the reference system o .
The observer B o registers: After time t T = t P = x P /v, brother A o changes into the
reference system
and moves behind his twin brother A with the velocity u > v.
This ‘changing of reference systems’ is the displayed event T in Fig. 17.3 with its
coordinates in o according to (184) and in
according to (188). Twin A is located
at x P during event T in o . From observer B o ’s point of view, brother A o needs
t u =
x P
u−v
from the point when he changed reference systems until the point where
he catches up with his brother A o . Once more, t u is the time measured from out
of o that brother A o spent in
. We will understand further down the reference
system that moves with the special velocity u according to (196), with respect to o ,
by
. The twin’s meeting is defined, if we have the case of a general velocity u, as
the event S(u). From both of the times in o
t T =
x P
v
,
t u =
x P
u − v
, u − v > 0 ,
⎫
⎬
⎭
(189)
and from the velocity u after the change of reference systems, the observer B o
calculates according to our Eq. (121) the hand setting of brother A o ’s clock U
A
o as
the coefficient of measure ˜
t S(u) = t T + t u
1 − u 2 /c 2
o and thus
˜
t S(u) =
x P
v
1 +
v
u − v
1 −
u 2
c 2
o
.
(190)
On the other hand, due to the fact that twin A constantly moves with the velocity v,
the observer B o calculates according the hand setting of brother A’s clock U
A as the
coefficient of measure t
S(u) = (t T + t u )
1 − v 2 /c 2
o and thus
t
S(u) =
x P
v
+
x P
u − v
1 −
v 2
c 2
o
=
x P
v
1 +
v
u − v
1 −
v 2
c 2
o
,
17 The Twin Paradox
The later therefore aged less during his journey (which should compensate him
for losing the farm)—stays young and drives fast.
This result is from twin A’s point of view of course self-evident, because it is
brother A o who moves. And it is the moving clock that goes behind. The velocity
with which A o departs from A and the then following arbitrary velocity chosen by
A o to return to twin A should make no difference to the sign of this effect, of course
from out of the view of A in the reference system
. However, what does this case
look like if it is observed from the reference system o , the reference system in
which our observer B o controls the personal clocks of both twin brothers?
We will accept that it is once again twin A who moves with the constant velocity
v with respect to the reference system o . Brother A o sits and watches, together
with observer B o who makes all of his measurements in the reference system o .
The observer B o registers: After time t T = t P = x P /v, brother A o changes into the
reference system
and moves behind his twin brother A with the velocity u > v.
This ‘changing of reference systems’ is the displayed event T in Fig. 17.3 with its
coordinates in o according to (184) and in
according to (188). Twin A is located
at x P during event T in o . From observer B o ’s point of view, brother A o needs
t u =
x P
u−v
from the point when he changed reference systems until the point where
he catches up with his brother A o . Once more, t u is the time measured from out
of o that brother A o spent in
. We will understand further down the reference
system that moves with the special velocity u according to (196), with respect to o ,
by
. The twin’s meeting is defined, if we have the case of a general velocity u, as
the event S(u). From both of the times in o
t T =
x P
v
,
t u =
x P
u − v
, u − v > 0 ,
⎫
⎬
⎭
(189)
and from the velocity u after the change of reference systems, the observer B o
calculates according to our Eq. (121) the hand setting of brother A o ’s clock U
A
o as
the coefficient of measure ˜
t S(u) = t T + t u
1 − u 2 /c 2
o and thus
˜
t S(u) =
x P
v
1 +
v
u − v
1 −
u 2
c 2
o
.
(190)
On the other hand, due to the fact that twin A constantly moves with the velocity v,
the observer B o calculates according the hand setting of brother A’s clock U
A as the
coefficient of measure t
S(u) = (t T + t u )
1 − v 2 /c 2
o and thus
t
S(u) =
x P
v
+
x P
u − v
1 −
v 2
c 2
o
=
x P
v
1 +
v
u − v
1 −
v 2
c 2
o
,