17 The Twin Paradox
185
We can insert this initial condition into Fig. 17.3; see Fig. 17.6. We still miss the
space coordinate x
P for the event P. We can determine this as follows: A coordinate
difference with the amount x P determined in o at one and the same time t T = t P is
the coefficient of measure of a length moving with the velocity u in o , therefore a
static length in
with the coefficient of measure x
P . Due to the Lorentz contraction
(113a), x
P =
x P
√
1−u 2 /c 2
o
, hence with the velocity u according to (197)
x
P = x P
c
2
o + v
2
c 2
o − v 2 .
The coordinate x
P is the coefficient of measure of the determined distance between the events T and P in
using the measuring-rod L
of
, whereby
L
= L o
1 − u 2 /c 2
o = L o
c
2
o +v
2
c 2
o −v 2 because of the Lorentz contraction (112), as shown
in Fig. 17.6.
At x
P we observe the clock U
P
t resting in
. In order to be able to determine
the hand setting of the clock U
P
t at event P, the time coordinate P in
, we need
to take into consideration that the events T and P occur simultaneously in o . In
Eq. (142a), see also Figs. 12.6 and 12.7, we managed to calculate with the help of
our elementary principle of relativity, the number of scale marks t
that the hand
of the clock at the position x o + x in o moving with the velocity v with respect
to o has to be moved forwards or backwards, as against the clock moving with the
velocity v at x o . We use this in our case where u replaces v, as well as x o = 0 and
x P instead of x. We now write t
= t
P − t
T for t
(our clock U
T
t in
already
shows the different from zero initial position t T = t
T ), thus
t
= t
P − t
T =
−x u/c
2
o
1 − u 2 /c 2
o
=
−x P u/c
2
o
1 − u 2 /c 2
o
.
(208)
With (196) and (197) for u, the following is valid
t
= −
x P
v
2v
2
c 2
o − v 2 .
Difference of the clock’s
U
P
t and U
T
t hand setting
in
(209)
We note that the difference t
= t
P − t
T of the hand settings of both clocks U
P
t
and U
T
t is of course independent of the arbitrary choice of initial condition. The
initial position does however determine the absolute hand setting. With t
T = x P /v
according to (207), we discover for the first case of the initial condition
t
P =
x P
v
−
x P
v
2v
2
c 2
o − v 2 =
x P
v
1 −
2v
2
c 2
o − v 2
.
Summarising this, we discover for the event P in
; see also Fig. 17.6,
185
We can insert this initial condition into Fig. 17.3; see Fig. 17.6. We still miss the
space coordinate x
P for the event P. We can determine this as follows: A coordinate
difference with the amount x P determined in o at one and the same time t T = t P is
the coefficient of measure of a length moving with the velocity u in o , therefore a
static length in
with the coefficient of measure x
P . Due to the Lorentz contraction
(113a), x
P =
x P
√
1−u 2 /c 2
o
, hence with the velocity u according to (197)
x
P = x P
c
2
o + v
2
c 2
o − v 2 .
The coordinate x
P is the coefficient of measure of the determined distance between the events T and P in
using the measuring-rod L
of
, whereby
L
= L o
1 − u 2 /c 2
o = L o
c
2
o +v
2
c 2
o −v 2 because of the Lorentz contraction (112), as shown
in Fig. 17.6.
At x
P we observe the clock U
P
t resting in
. In order to be able to determine
the hand setting of the clock U
P
t at event P, the time coordinate P in
, we need
to take into consideration that the events T and P occur simultaneously in o . In
Eq. (142a), see also Figs. 12.6 and 12.7, we managed to calculate with the help of
our elementary principle of relativity, the number of scale marks t
that the hand
of the clock at the position x o + x in o moving with the velocity v with respect
to o has to be moved forwards or backwards, as against the clock moving with the
velocity v at x o . We use this in our case where u replaces v, as well as x o = 0 and
x P instead of x. We now write t
= t
P − t
T for t
(our clock U
T
t in
already
shows the different from zero initial position t T = t
T ), thus
t
= t
P − t
T =
−x u/c
2
o
1 − u 2 /c 2
o
=
−x P u/c
2
o
1 − u 2 /c 2
o
.
(208)
With (196) and (197) for u, the following is valid
t
= −
x P
v
2v
2
c 2
o − v 2 .
Difference of the clock’s
U
P
t and U
T
t hand setting
in
(209)
We note that the difference t
= t
P − t
T of the hand settings of both clocks U
P
t
and U
T
t is of course independent of the arbitrary choice of initial condition. The
initial position does however determine the absolute hand setting. With t
T = x P /v
according to (207), we discover for the first case of the initial condition
t
P =
x P
v
−
x P
v
2v
2
c 2
o − v 2 =
x P
v
1 −
2v
2
c 2
o − v 2
.
Summarising this, we discover for the event P in
; see also Fig. 17.6,