184
17 The Twin Paradox
What is important here is that the events T and P are only simultaneous in o .
Brother A o can only state, as long as he is in the reference system o that he knows
the setting of twin A’s clock U
A at the time he boards the train according to (206).
As soon as brother A o is in the reference system
—and this is where the snare
is—his statement is incorrect! The statement ‘at the time he boards the train’ only
means when all clocks in o show t = x P /v, thus also the o clock at x = x P , only
then does A’s clock U
A moving past this exact location with the velocity v shows
the setting
x P
v
1 − v 2 /c 2
o .
Let us remind ourselves that: We now have to determine the rule by which the
clocks in
are started. By this, we have defined what it means when we state:
The clocks in different positions in
run synchronically. In Chap. 12, we used our
elementary principle of relativity as the rule by which we synchronised clocks. Let
us take up again our quotation on p. 119. In 1898, seven years before the discovery of
Einstein’s Special Theory of Relativity, Poincaré [; ] wrote: ‘Il est difficile de séparer
le probl` eme qualitative de la simultanéité du probl` eme quantitative de la mesure du
temps; soit qu’on se serve d’un chronom` etre, soit qu’on ait `
a tenir compte d’une
vitesse de transmission, comme celle de la lumi` ere, car on ne saurait mesurer une
pareille vitesse sans mesurer un temps. · · · Nous n’avons pas l’intuition directe de
la simultanéité, par plus que selle de l’égalité de deux durées. Si nous croyons avoir
cette intuition, c’est une illusion”.
3
After changing to the reference system
, brother A o has the velocity u with
respect to o and the velocity v with respect to
. We will also position a neutral
observer B
in the reference system
, and we equip him with a clock U
T
t . (The
other clocks in the reference system
as well are distinguished with the index
‘t ’, which refers to the ‘train’). The observer B
should be noticed in the reference
system o at the event T , in other words, B
, who has the velocity u in o , has the
coordinate x T = 0 at time t T = x P /v in o .
We still have one free initial condition. In other words, we can pick out any clock
in
and arbitrarily start it. All other clocks then depend on it. We can also freely
choose an initial point for the counting of the space coordinates in
. We wish to
compare two cases that have equal rights with one another.
1. We firstly determine that: The static clock in U
T
t in
takes over the hand setting
of brother A o ’s clock U
A
o at event T . In other words, the hand setting of observer
B
’s clock U
T
t is set to t
T = t T = x P /v during event T . The space coordinate of the
event T in
should be x
T = 0. We therefore choose the first case for the initial
condition in
according to:
: T
x
T = 0, t
T =
x P
v
.
Initial condition in
First case
(207)
3 It is difficult to separate the qualitative problem of simultaneity from the quantitative problem of
the measurement of time; no matter whether a chronometer is used, or whether account must be
taken of a velocity of transmission, as that of light, because such a velocity could not be measured
without measuring a time.· · ·. We have not a direct intuition of simultaneity, nor of the equality
of two durations. If we think we have this intuition, this is an illusion’.
17 The Twin Paradox
What is important here is that the events T and P are only simultaneous in o .
Brother A o can only state, as long as he is in the reference system o that he knows
the setting of twin A’s clock U
A at the time he boards the train according to (206).
As soon as brother A o is in the reference system
—and this is where the snare
is—his statement is incorrect! The statement ‘at the time he boards the train’ only
means when all clocks in o show t = x P /v, thus also the o clock at x = x P , only
then does A’s clock U
A moving past this exact location with the velocity v shows
the setting
x P
v
1 − v 2 /c 2
o .
Let us remind ourselves that: We now have to determine the rule by which the
clocks in
are started. By this, we have defined what it means when we state:
The clocks in different positions in
run synchronically. In Chap. 12, we used our
elementary principle of relativity as the rule by which we synchronised clocks. Let
us take up again our quotation on p. 119. In 1898, seven years before the discovery of
Einstein’s Special Theory of Relativity, Poincaré [; ] wrote: ‘Il est difficile de séparer
le probl` eme qualitative de la simultanéité du probl` eme quantitative de la mesure du
temps; soit qu’on se serve d’un chronom` etre, soit qu’on ait `
a tenir compte d’une
vitesse de transmission, comme celle de la lumi` ere, car on ne saurait mesurer une
pareille vitesse sans mesurer un temps. · · · Nous n’avons pas l’intuition directe de
la simultanéité, par plus que selle de l’égalité de deux durées. Si nous croyons avoir
cette intuition, c’est une illusion”.
3
After changing to the reference system
, brother A o has the velocity u with
respect to o and the velocity v with respect to
. We will also position a neutral
observer B
in the reference system
, and we equip him with a clock U
T
t . (The
other clocks in the reference system
as well are distinguished with the index
‘t ’, which refers to the ‘train’). The observer B
should be noticed in the reference
system o at the event T , in other words, B
, who has the velocity u in o , has the
coordinate x T = 0 at time t T = x P /v in o .
We still have one free initial condition. In other words, we can pick out any clock
in
and arbitrarily start it. All other clocks then depend on it. We can also freely
choose an initial point for the counting of the space coordinates in
. We wish to
compare two cases that have equal rights with one another.
1. We firstly determine that: The static clock in U
T
t in
takes over the hand setting
of brother A o ’s clock U
A
o at event T . In other words, the hand setting of observer
B
’s clock U
T
t is set to t
T = t T = x P /v during event T . The space coordinate of the
event T in
should be x
T = 0. We therefore choose the first case for the initial
condition in
according to:
: T
x
T = 0, t
T =
x P
v
.
Initial condition in
First case
(207)
3 It is difficult to separate the qualitative problem of simultaneity from the quantitative problem of
the measurement of time; no matter whether a chronometer is used, or whether account must be
taken of a velocity of transmission, as that of light, because such a velocity could not be measured
without measuring a time.· · ·. We have not a direct intuition of simultaneity, nor of the equality
of two durations. If we think we have this intuition, this is an illusion’.