168
17 The Twin Paradox
Fig. 17.1 Event O. The departure of the twins from the common coordinate origin. Brother A o has
the clock U A
o , and the twin A has the clock U A with him. Brother A o observes that the measuringrod L of his twin A is shorter than his measuring-rod L o by the factor γ. We once again consider
the case v = 0, 8 c o , thus γ = 0, 6, which we have depicted as L = 0, 6 L o . We have not plotted
the contraction in length that the twin A measures. He determines the same factor γ for the moving
length L v = γ L of the, from his point of view, moving measuring rod L o ; see Fig.15.2
older each twin has become since their departure. The space and time statements of an
event are once again shown in o with the help of unprimed coordinates (x, t) and in
by using primed coordinates (x
, t
). We recapitulate as follows: The coordinates
are always the coefficients of measure of lengths and time. The hands of the clocks
show the coefficients of the measure of time. These following stated coordinates of
time can thus be immediately read from the clocks.
In order to suppress as many objections as possible concerning the twin paradox,
we will discuss this problem in detail from various viewpoints. A simple, special
solution of this paradox can be found on the pp. 173–175.
Brother A o reasons in his reference system o : My twin brother A with his clock
U
A is moving with the velocity v. After the time t p = x P /v according to my clock
U
A
o , my twin has arrived at the location x P in o . We will call this the event P with
the coordinates in o according to
o : P
x P , t P =
x P
v
.
(178)
The hands of the clock U
A of twin A are thus positioned, according to our Eq. (121)
at the time of t
P = t P
1 − v 2 /c 2
o . In the reference system
, event P thus has the
coordinates
: P
x
P = 0, t
P =
x P
v
1 −
v 2
c 2
o
.
(179)
Thus, when twin A compares the time t
P shown on his clock to the time t P shown
on the clock in o opposite from him, let us call this clock U
P
o , and he notices that
17 The Twin Paradox
Fig. 17.1 Event O. The departure of the twins from the common coordinate origin. Brother A o has
the clock U A
o , and the twin A has the clock U A with him. Brother A o observes that the measuringrod L of his twin A is shorter than his measuring-rod L o by the factor γ. We once again consider
the case v = 0, 8 c o , thus γ = 0, 6, which we have depicted as L = 0, 6 L o . We have not plotted
the contraction in length that the twin A measures. He determines the same factor γ for the moving
length L v = γ L of the, from his point of view, moving measuring rod L o ; see Fig.15.2
older each twin has become since their departure. The space and time statements of an
event are once again shown in o with the help of unprimed coordinates (x, t) and in
by using primed coordinates (x
, t
). We recapitulate as follows: The coordinates
are always the coefficients of measure of lengths and time. The hands of the clocks
show the coefficients of the measure of time. These following stated coordinates of
time can thus be immediately read from the clocks.
In order to suppress as many objections as possible concerning the twin paradox,
we will discuss this problem in detail from various viewpoints. A simple, special
solution of this paradox can be found on the pp. 173–175.
Brother A o reasons in his reference system o : My twin brother A with his clock
U
A is moving with the velocity v. After the time t p = x P /v according to my clock
U
A
o , my twin has arrived at the location x P in o . We will call this the event P with
the coordinates in o according to
o : P
x P , t P =
x P
v
.
(178)
The hands of the clock U
A of twin A are thus positioned, according to our Eq. (121)
at the time of t
P = t P
1 − v 2 /c 2
o . In the reference system
, event P thus has the
coordinates
: P
x
P = 0, t
P =
x P
v
1 −
v 2
c 2
o
.
(179)
Thus, when twin A compares the time t
P shown on his clock to the time t P shown
on the clock in o opposite from him, let us call this clock U
P
o , and he notices that