17 The Twin Paradox
171
that his, the twin’s A version is the more valid. He should return as fast as possible
in order to hand over the farm to the older brother. Time is pressing. How does twin
A reply? The equation for the time delay of clocks only contains the square of the
journey’s velocity v. If he, twin A, moves with −v with respect to brother A o , then
A o moves with +v with respect to A. Then, A o ’s clock U
A
o that already went behind
on the first leg of the journey (with respect to A) should still go behind. Following
from this, when the brothers meet, the clock U
A
o should still go behind the A’s clock
U
A and not vice versa. However, one of both cases cannot apply. The hand setting
of a clock is a fact that cannot be ignored.
One single hand in two different positions—that would be paradox!
In fact, somewhere along the way an error has occurred, an error that we want
to find. Correct without doubt, according to (121) is that it makes no difference
for the time delay of twin A’s moving clock whether he travels away from brother
A o with the velocity +v, or if he moves towards him with the velocity −v. Fact is
that when they meet, the hand of U
A goes behind the hand of U
A
o by the amount
2
x P
v
1 − v 2 /c 2
o . Thus, the mistake must lie in twin A’s argumentation.
If we examine Eqs. (121) and (162) for the time dilatation of moving clocks, then
we see that these depend on the square of velocity v. Let us go back one step and
consider the regulations for synchronising clocks in a reference system
moving
with the velocity +v with respect to a reference system o . This is Eq. (142) as shown
in Fig.12.7. This synchronisation completely depends on the direction of the velocity
v ! Reversing the velocity v forces the clocks to be set in the reverse direction. Let
us recapitulate as follows:
The time dilatation of a moving clock is solely dependent on the square of its velocity. The
synchronisation regulation of a clock changes its sign when the direction of velocity changes.
This asymmetry in the synchronisation of clocks plays a decisive role in the
clarification of the twin paradox.
If twin A wants to return to his brother A o , he must leave his reference system
and enter a new reference system, let us call it
− , whose clocks are synchronised
so that
− has the velocity −v with respect to o and that this synchronisation
contrarotates the one in the reference system
used by twin A in his statements
concerning the time flow. For twin A, it is now obliging to use the new synchronisation
regulation for his time comparison with o . For brother A o , who only needs the time
dilatation of U
A , this new synchronisation plays no role, because we assumed that
twin A would be returning with the velocity −v.
The situation is reversed if twin A remains in his reference system
and his
brother A o leaves his reference system o in order to enter a new reference system,
let us say
and rushes after his twin A, so that twin A registers the velocity of A o
travelling towards him as +v. The velocity +v with respect to
is now decisive for
the synchronisation of clocks in
, and this synchronisation contrarotates the one in
the reference system o , which was decisive for brother A o ’s statements concerning
the time span. For brother A o , the new synchronisation regulation is valid when
comparing time with
. Now, both brothers meet in
at x
= 0. If the reverse in
171
that his, the twin’s A version is the more valid. He should return as fast as possible
in order to hand over the farm to the older brother. Time is pressing. How does twin
A reply? The equation for the time delay of clocks only contains the square of the
journey’s velocity v. If he, twin A, moves with −v with respect to brother A o , then
A o moves with +v with respect to A. Then, A o ’s clock U
A
o that already went behind
on the first leg of the journey (with respect to A) should still go behind. Following
from this, when the brothers meet, the clock U
A
o should still go behind the A’s clock
U
A and not vice versa. However, one of both cases cannot apply. The hand setting
of a clock is a fact that cannot be ignored.
One single hand in two different positions—that would be paradox!
In fact, somewhere along the way an error has occurred, an error that we want
to find. Correct without doubt, according to (121) is that it makes no difference
for the time delay of twin A’s moving clock whether he travels away from brother
A o with the velocity +v, or if he moves towards him with the velocity −v. Fact is
that when they meet, the hand of U
A goes behind the hand of U
A
o by the amount
2
x P
v
1 − v 2 /c 2
o . Thus, the mistake must lie in twin A’s argumentation.
If we examine Eqs. (121) and (162) for the time dilatation of moving clocks, then
we see that these depend on the square of velocity v. Let us go back one step and
consider the regulations for synchronising clocks in a reference system
moving
with the velocity +v with respect to a reference system o . This is Eq. (142) as shown
in Fig.12.7. This synchronisation completely depends on the direction of the velocity
v ! Reversing the velocity v forces the clocks to be set in the reverse direction. Let
us recapitulate as follows:
The time dilatation of a moving clock is solely dependent on the square of its velocity. The
synchronisation regulation of a clock changes its sign when the direction of velocity changes.
This asymmetry in the synchronisation of clocks plays a decisive role in the
clarification of the twin paradox.
If twin A wants to return to his brother A o , he must leave his reference system
and enter a new reference system, let us call it
− , whose clocks are synchronised
so that
− has the velocity −v with respect to o and that this synchronisation
contrarotates the one in the reference system
used by twin A in his statements
concerning the time flow. For twin A, it is now obliging to use the new synchronisation
regulation for his time comparison with o . For brother A o , who only needs the time
dilatation of U
A , this new synchronisation plays no role, because we assumed that
twin A would be returning with the velocity −v.
The situation is reversed if twin A remains in his reference system
and his
brother A o leaves his reference system o in order to enter a new reference system,
let us say
and rushes after his twin A, so that twin A registers the velocity of A o
travelling towards him as +v. The velocity +v with respect to
is now decisive for
the synchronisation of clocks in
, and this synchronisation contrarotates the one in
the reference system o , which was decisive for brother A o ’s statements concerning
the time span. For brother A o , the new synchronisation regulation is valid when
comparing time with
. Now, both brothers meet in
at x
= 0. If the reverse in