17 The Twin Paradox
193
respect to o , for which we calculated, according to (198) the duration of stay
in o according to t v =
x P
v
c
2
o +v
2
c 2
o −v 2 . Applying (197), it follows immediately ˜ t v =
t v ·
1 − u 2 /c 2
o .
It thus remains as it was, during the complete journey the hand of brother A o ’s,
the one who rushes back towards twin A, clock U
A
o only moves forwards by ˜
t S(v) =
2x P /v,
A o : ˜
t S(v) = 2
x P
v
.
Hand setting of the clock U
A
o
at the point of reunion
(214)
It therefore goes behind twin A’s clock U
A by the factor
1 − v 2 /c 2
o , which
shows t
S(v) =
2x P /v
√
1−v 2 /c 2
o
at the reunion.
The returning brother, or the brother who rushes after the other respectively is the younger.
Brother A o had manipulated his clock. He has been unmasked. The paradox has
been solved; see Fig. 17.9.
The geometric versified reader will anticipate that these connections
virtually demand for geometric description. For this, we especially refer the reader
to Liebscher’s [] presentation of Special Relativity.
It is fascinating to imagine how all these processes occur inside of a crystal,
how the oscillating breathers move through the crystal with varying velocities and
frequencies, how various breather groups with the same velocity can be formed into
a reference system, how these are synchronised with respect to one another, such
that the time displayed in one system behaves exactly as shown in Fig.12.7, to that
of another breather system moving through the crystal with a different velocity. The
end result is that every single phase of the twin paradox can be illustrated using an
oscillating breather moving through the crystal.
4
We outside experimenters have a privileged position with respect to that of the
internal observers, because we can determine the motion state of the crystal as a
whole by using our outside simultaneity, as defined by the speed of light. Although
our ‘outside clocks’ and ‘outside measuring-rods’, the clocks and measuring-rods
that every physicist uses in his measurements, underlie the laws of time dilatation
and length contraction, we use the speed of light c L , whereas the internal observer
uses the critical velocity c o . As explained in Chap. 11, the physically identical clocks,
which only move relatively with respect to one another, used by the internal observers
inside of the crystal are from the point of view of the outside observers not physically
identical clocks. Therefore, for the outside observer, the whole complicated and
strenuous internal procedure of synchronising the clocks seems to be most strange and
at first unnecessary. However, the internal time processes registered by the internal
4 Notice that all formulas for the coordinates of the various events in different frames o , , , , , . . .
directly can be controlled with the help of Lorentz transformation (151).
193
respect to o , for which we calculated, according to (198) the duration of stay
in o according to t v =
x P
v
c
2
o +v
2
c 2
o −v 2 . Applying (197), it follows immediately ˜ t v =
t v ·
1 − u 2 /c 2
o .
It thus remains as it was, during the complete journey the hand of brother A o ’s,
the one who rushes back towards twin A, clock U
A
o only moves forwards by ˜
t S(v) =
2x P /v,
A o : ˜
t S(v) = 2
x P
v
.
Hand setting of the clock U
A
o
at the point of reunion
(214)
It therefore goes behind twin A’s clock U
A by the factor
1 − v 2 /c 2
o , which
shows t
S(v) =
2x P /v
√
1−v 2 /c 2
o
at the reunion.
The returning brother, or the brother who rushes after the other respectively is the younger.
Brother A o had manipulated his clock. He has been unmasked. The paradox has
been solved; see Fig. 17.9.
The geometric versified reader will anticipate that these connections
virtually demand for geometric description. For this, we especially refer the reader
to Liebscher’s [] presentation of Special Relativity.
It is fascinating to imagine how all these processes occur inside of a crystal,
how the oscillating breathers move through the crystal with varying velocities and
frequencies, how various breather groups with the same velocity can be formed into
a reference system, how these are synchronised with respect to one another, such
that the time displayed in one system behaves exactly as shown in Fig.12.7, to that
of another breather system moving through the crystal with a different velocity. The
end result is that every single phase of the twin paradox can be illustrated using an
oscillating breather moving through the crystal.
4
We outside experimenters have a privileged position with respect to that of the
internal observers, because we can determine the motion state of the crystal as a
whole by using our outside simultaneity, as defined by the speed of light. Although
our ‘outside clocks’ and ‘outside measuring-rods’, the clocks and measuring-rods
that every physicist uses in his measurements, underlie the laws of time dilatation
and length contraction, we use the speed of light c L , whereas the internal observer
uses the critical velocity c o . As explained in Chap. 11, the physically identical clocks,
which only move relatively with respect to one another, used by the internal observers
inside of the crystal are from the point of view of the outside observers not physically
identical clocks. Therefore, for the outside observer, the whole complicated and
strenuous internal procedure of synchronising the clocks seems to be most strange and
at first unnecessary. However, the internal time processes registered by the internal
4 Notice that all formulas for the coordinates of the various events in different frames o , , , , , . . .
directly can be controlled with the help of Lorentz transformation (151).