108
11 A Clock Paradox
Time dilatation is the ‘experimentum crucis’ of Special Relativity. The physical
constitution or the technical construction of clocks is not found in the equation for
time dilatation. We can thus use our breather clocks in our test for the dilatation of
time in the same way as we examined the pace of these clocks in the last chapter.
Here, we discover that the hand of a moving breather clock went behind the hands of
the static breather clocks according to Eq. (121) t
= t
1 − v 2 /c 2
o . Now factually,
this is a dilatation of time as seen in the Einsteinian theory of relativity, but the
numerical value is completely different. In our equation, it is not the speed of light
c L , but the characteristic velocity c o of the sine-Gordon equation that is included.
This velocity c o lies in the region of the velocities of sound and is thus smaller
than the speed of light by many powers of ten, c o c L . Einstein’s Special Relativity
states, however, that only the speed of light alone can stand in the equation for time
dilatation for all clocks. This is the focal point of the whole theory! Have we just
shown that Einstein’s Special Theory of Relativity is not universally applicable or
maybe even incorrect?—By no means.
Where can we find the solution to this paradox: The crucial statement we used
to introduce the description of the dilatation of time was ‘we construct a number
of identical clocks’. The static breather clocks positioned in the inertial system o
are oscillating lines q
III
o where the geometrical centres of inertia rest relative to the
crystal. However, our moving breather clock is an oscillating line q
III , where the
geometrical centre of inertia has the velocity v relative to the crystal. The observer
of the Einsteinian theory of relativity, the ‘outside observer’, wanting to use crystals
in the construction of clocks examines the crystal with its oscillating breather lines
as a whole. An outside observer stationary in o may introduce the oscillating lines
q
III
o inside a crystal, whose centres of inertia rest relative to the crystal, in order to
measure time. Moving clocks of this construction are for this observer always of
such a kind that the whole crystal moves with its oscillating lines. The construction
rules of such a clock for the outside observer include that the centres of inertia of the
oscillating lines remain motionless with respect to the crystal. This condition must
remain fulfilled when clocks move. These oscillating lines q
III with their centres of
inertia have the velocity v with respect to the crystal, are for this observer completely
different clocks. Such clocks could also have been introduced in order to measure
time. The observer of the Einsteinian Special Theory of Relativity finds it normal
that both types of clocks have different periods of oscillation. In order to compare
these, they would have to be gauged accordingly just as an oscillating quartz would
have to oscillate more times in a second than the spring of a pocket watch. Thus, the
observer would not arrive at a contradiction in his statements on Special Relativity. If
the observer uses the static breather clock inside the crystal, so that the clock moves
together with the crystal, he finds after comparing the hand setting ˜
t of a breather
clock moving together with the crystal with the hand settings t of such breather
(Footnote 1 continued)
this would need a complete new line of thought. This new line of thought would also play no role
in our needs for solids. (For a pendulum clock moving with the velocity v away from earth, the
influence of earth’s gravity would decrease; however, the influence from the moon could increase).
11 A Clock Paradox
Time dilatation is the ‘experimentum crucis’ of Special Relativity. The physical
constitution or the technical construction of clocks is not found in the equation for
time dilatation. We can thus use our breather clocks in our test for the dilatation of
time in the same way as we examined the pace of these clocks in the last chapter.
Here, we discover that the hand of a moving breather clock went behind the hands of
the static breather clocks according to Eq. (121) t
= t
1 − v 2 /c 2
o . Now factually,
this is a dilatation of time as seen in the Einsteinian theory of relativity, but the
numerical value is completely different. In our equation, it is not the speed of light
c L , but the characteristic velocity c o of the sine-Gordon equation that is included.
This velocity c o lies in the region of the velocities of sound and is thus smaller
than the speed of light by many powers of ten, c o c L . Einstein’s Special Relativity
states, however, that only the speed of light alone can stand in the equation for time
dilatation for all clocks. This is the focal point of the whole theory! Have we just
shown that Einstein’s Special Theory of Relativity is not universally applicable or
maybe even incorrect?—By no means.
Where can we find the solution to this paradox: The crucial statement we used
to introduce the description of the dilatation of time was ‘we construct a number
of identical clocks’. The static breather clocks positioned in the inertial system o
are oscillating lines q
III
o where the geometrical centres of inertia rest relative to the
crystal. However, our moving breather clock is an oscillating line q
III , where the
geometrical centre of inertia has the velocity v relative to the crystal. The observer
of the Einsteinian theory of relativity, the ‘outside observer’, wanting to use crystals
in the construction of clocks examines the crystal with its oscillating breather lines
as a whole. An outside observer stationary in o may introduce the oscillating lines
q
III
o inside a crystal, whose centres of inertia rest relative to the crystal, in order to
measure time. Moving clocks of this construction are for this observer always of
such a kind that the whole crystal moves with its oscillating lines. The construction
rules of such a clock for the outside observer include that the centres of inertia of the
oscillating lines remain motionless with respect to the crystal. This condition must
remain fulfilled when clocks move. These oscillating lines q
III with their centres of
inertia have the velocity v with respect to the crystal, are for this observer completely
different clocks. Such clocks could also have been introduced in order to measure
time. The observer of the Einsteinian Special Theory of Relativity finds it normal
that both types of clocks have different periods of oscillation. In order to compare
these, they would have to be gauged accordingly just as an oscillating quartz would
have to oscillate more times in a second than the spring of a pocket watch. Thus, the
observer would not arrive at a contradiction in his statements on Special Relativity. If
the observer uses the static breather clock inside the crystal, so that the clock moves
together with the crystal, he finds after comparing the hand setting ˜
t of a breather
clock moving together with the crystal with the hand settings t of such breather
(Footnote 1 continued)
this would need a complete new line of thought. This new line of thought would also play no role
in our needs for solids. (For a pendulum clock moving with the velocity v away from earth, the
influence of earth’s gravity would decrease; however, the influence from the moon could increase).