17 The Twin Paradox
189
Due to time dilatation of the clock U
A moving with the velocity v with respect to
, the hand of the clock moves forward by t
PV = t
PV
1 − v 2 /c 2
o , so that they take
up the position t
V = t
P + t
PV at event V , thus with t
P , according to (206),
t
V =
x P
v
1 −
v 2
c 2
o
+
x P
v
2v
2
c 2
o − v 2
1 −
v 2
c 2
o
=
x P
v
1 −
v 2
c 2
o
1 +
2v
2
c 2
o − v 2
,
=
x P
v
1 −
v 2
c 2
o
c
2
o + v
2
c 2
o − v 2 =
x P
v
1 −
v 2
c 2
o
1 + v
2
/c
2
o
1 − v 2 /c 2
o
=
x P
v
1 + v
2
/c
2
o
1 − v 2 /c 2
o
.
This is therefore the hand setting of twin A’s clock U
A at event T in
, as registered
by brother A o after transferring into this reference system; see also Fig. 17.8,
A o : t
V =
x P
v
1 + v
2
/c
2
o
1 − v 2 /c 2
o
.
Hand setting of the clock U
A
at the event T in
(217)
Up to the reunion of the brothers at event S(v), the hand of brother A o ’s clock U
A
o
moved forward, according to (213) by ˜ t v = x P /v scale marks. The hand of the clock
U
A therefore only moved forward, due to time dilatation, by t
v = ˜ t v
1 − v 2 /c 2
o
and takes up the position at event S(v)
t
S(v) =
x P
v
1 + v
2
/c
2
o
1 − v 2 /c 2
o
+
x P
v
1 −
v
2
c 2
o
=
x P
v
1
1 − v 2 /c 2
o
⎛
⎝ 1 +
v
2
c 2
o
+
1 −
v
2
c 2
o
1 −
v
2
c 2
o
⎞
⎠ =
x P
v
2
1 − v 2 /c 2
o
.
Brother A o in fact observes
A o : t
S(v) =
2x P /v
1 − v 2 /c 2
o
,
Hand setting of the clock U
A
at the point of reunion
(218)
which is completely in accordance with the statement (204) of twin A, as well as the
statement (200) made by observer B o . Only the hand of brother A o ’s personal clock
U
A
o goes behind that of twin A’s personal clock U
A , just as we awaited and expected.
The paradox is there clarified. We now also want to see exactly where the trap
is set for a second case of the initial condition of space and time measurement in
the reference system
. In order to enable brother A o , even after he has changed
reference systems, to measure the passed time on twin A’s clock U
A , we choose for
this second case the following initial condition:
2. The clock U
P
t resting in
takes over the hand setting of the clock U
A
o at event
P. This means that the hand of the clock U
P
t is set to t
P = t P = x P /v at event P. We
also set the initial point for the counting of the space coordinates in
to x
P = 0;
189
Due to time dilatation of the clock U
A moving with the velocity v with respect to
, the hand of the clock moves forward by t
PV = t
PV
1 − v 2 /c 2
o , so that they take
up the position t
V = t
P + t
PV at event V , thus with t
P , according to (206),
t
V =
x P
v
1 −
v 2
c 2
o
+
x P
v
2v
2
c 2
o − v 2
1 −
v 2
c 2
o
=
x P
v
1 −
v 2
c 2
o
1 +
2v
2
c 2
o − v 2
,
=
x P
v
1 −
v 2
c 2
o
c
2
o + v
2
c 2
o − v 2 =
x P
v
1 −
v 2
c 2
o
1 + v
2
/c
2
o
1 − v 2 /c 2
o
=
x P
v
1 + v
2
/c
2
o
1 − v 2 /c 2
o
.
This is therefore the hand setting of twin A’s clock U
A at event T in
, as registered
by brother A o after transferring into this reference system; see also Fig. 17.8,
A o : t
V =
x P
v
1 + v
2
/c
2
o
1 − v 2 /c 2
o
.
Hand setting of the clock U
A
at the event T in
(217)
Up to the reunion of the brothers at event S(v), the hand of brother A o ’s clock U
A
o
moved forward, according to (213) by ˜ t v = x P /v scale marks. The hand of the clock
U
A therefore only moved forward, due to time dilatation, by t
v = ˜ t v
1 − v 2 /c 2
o
and takes up the position at event S(v)
t
S(v) =
x P
v
1 + v
2
/c
2
o
1 − v 2 /c 2
o
+
x P
v
1 −
v
2
c 2
o
=
x P
v
1
1 − v 2 /c 2
o
⎛
⎝ 1 +
v
2
c 2
o
+
1 −
v
2
c 2
o
1 −
v
2
c 2
o
⎞
⎠ =
x P
v
2
1 − v 2 /c 2
o
.
Brother A o in fact observes
A o : t
S(v) =
2x P /v
1 − v 2 /c 2
o
,
Hand setting of the clock U
A
at the point of reunion
(218)
which is completely in accordance with the statement (204) of twin A, as well as the
statement (200) made by observer B o . Only the hand of brother A o ’s personal clock
U
A
o goes behind that of twin A’s personal clock U
A , just as we awaited and expected.
The paradox is there clarified. We now also want to see exactly where the trap
is set for a second case of the initial condition of space and time measurement in
the reference system
. In order to enable brother A o , even after he has changed
reference systems, to measure the passed time on twin A’s clock U
A , we choose for
this second case the following initial condition:
2. The clock U
P
t resting in
takes over the hand setting of the clock U
A
o at event
P. This means that the hand of the clock U
P
t is set to t
P = t P = x P /v at event P. We
also set the initial point for the counting of the space coordinates in
to x
P = 0;