188
17 The Twin Paradox
or, respectively, for the coordinates of event V
: V
x
V = x P , t
V =
x P
v
.
First case
(212)
Due to the velocity −v of twin A with respect to brother A o , the hand of brother A o ’s
clock U
A
o moves forward by the value ˜ t v when he steps on to the train,
˜ t v =
x P
v
.
At the clock U
A
o passed time
up to the point of reunion
(213)
This results, according to brother A o , together with (205) in a hand setting ˜
t S(v) =
˜
t T + ˜ t v for U
A
o at event S(v), the reunion of the brothers, thus,
A o : ˜
t S(v) = 2
x P
v
,
Hand setting of the clock U
A
o
at the point of reunion
(214)
corresponding to the results of not only observer B o according to (201), but also twin
A according to (203).
How does brother A o evaluate the passed time of twin A’s clock U
A ?
Due to time dilatation, the hand of twin A’s clock U
A moves forward by t
v
during time ˜ t v = x P /v according to
t
v =
x P
v
1 −
v 2
c 2
o
.
(215)
This is therefore the duration of brother A o with his clock U
A
o in the reference system
as measured from the reference system
.
What is however the position of clock’s U
A hand at the beginning of this part
of the journey, in other words at event V which has the coordinates x
V = x P and
t
V = x P /v in
? The paradox arises if one simply uses the statement (206) for this
hand setting that brother A o made before he jumped on to the train. This statement
however is incorrect, because the events P and V must not be confused! We need
the hand setting of the clock U
A at event V ! If we used the hand setting of U
A at
event P, according to (206), then we would, together with t
v , receive a journey
time of
2x P
v
1 − v 2 /c 2
o , resulting in the fact that twin A would be the younger. This
calculation is not allowable, as we have now seen, and the also result contradicts the
observations made by twin A on his own clock, according to (204), as well as those
made by observer B o , according to (200).
In
, between the events P and T (event T in
being simultaneous with event
V ) the following time passes, according to (209)
t
PT := t
T − t
P = t
V − t
P := t
PV = −t
=
x P
v
2v
2
c 2
o − v 2 .
(216)
17 The Twin Paradox
or, respectively, for the coordinates of event V
: V
x
V = x P , t
V =
x P
v
.
First case
(212)
Due to the velocity −v of twin A with respect to brother A o , the hand of brother A o ’s
clock U
A
o moves forward by the value ˜ t v when he steps on to the train,
˜ t v =
x P
v
.
At the clock U
A
o passed time
up to the point of reunion
(213)
This results, according to brother A o , together with (205) in a hand setting ˜
t S(v) =
˜
t T + ˜ t v for U
A
o at event S(v), the reunion of the brothers, thus,
A o : ˜
t S(v) = 2
x P
v
,
Hand setting of the clock U
A
o
at the point of reunion
(214)
corresponding to the results of not only observer B o according to (201), but also twin
A according to (203).
How does brother A o evaluate the passed time of twin A’s clock U
A ?
Due to time dilatation, the hand of twin A’s clock U
A moves forward by t
v
during time ˜ t v = x P /v according to
t
v =
x P
v
1 −
v 2
c 2
o
.
(215)
This is therefore the duration of brother A o with his clock U
A
o in the reference system
as measured from the reference system
.
What is however the position of clock’s U
A hand at the beginning of this part
of the journey, in other words at event V which has the coordinates x
V = x P and
t
V = x P /v in
? The paradox arises if one simply uses the statement (206) for this
hand setting that brother A o made before he jumped on to the train. This statement
however is incorrect, because the events P and V must not be confused! We need
the hand setting of the clock U
A at event V ! If we used the hand setting of U
A at
event P, according to (206), then we would, together with t
v , receive a journey
time of
2x P
v
1 − v 2 /c 2
o , resulting in the fact that twin A would be the younger. This
calculation is not allowable, as we have now seen, and the also result contradicts the
observations made by twin A on his own clock, according to (204), as well as those
made by observer B o , according to (200).
In
, between the events P and T (event T in
being simultaneous with event
V ) the following time passes, according to (209)
t
PT := t
T − t
P = t
V − t
P := t
PV = −t
=
x P
v
2v
2
c 2
o − v 2 .
(216)