10 Measuring-Rods and Clocks in Motion
93
then measure the time of flight, as we defined above, of an object between A and
B. This is not the problem. As long as we as an observer stay in one and the same
inertial system with our crystal, no complications should arise.
However, what happens if we sit on the moving object and observe the crystal, if
we then wish to synchronise clocks and use our measuring-rod from our position on
the moving object? What happens if we compare the synchronised clocks and the
measuring-rods on the moving object with those of the observer at rest with respect
to the crystal? See Fig. 33. What velocity does the observer on the flying object use to
synchronise his clocks? Does he too use the sound velocity? But the sound velocity
changes if the observer moves with respect to the crystal, and it even becomes dependent upon the direction of motion! There are even several sound velocities inside of a
crystal. Which of these should we use? We will not found our further considerations
on sound velocity, the critical velocity for a propagation of an elastic deformation. In
1905, this was with the discovery of the Special Theory of Relativity by A. Einstein
completely different. In those days, the observer independent, untouchable, infallible
and unbelievable constancy of the speed of light in a vacuum c L , which represents
the signal velocity of an arbitrary electromagnetic excitation, was made to the infallible principle of the resulting theory, out of which everything else was consequently
deduced, cf. Einstein’s axioms quoted in Chap. 2, cf. also Chap. 16.
We do not have such a distinguished maximum velocity inside of a crystal—at
least not yet. We do however have something else. We know far more about our
instruments of measurement, our measuring-rods and clocks. They are solutions of
a physical equation, the sine-Gordon equation. For the mysterious question in the
Special Theory of Relativity, what happens with the length of the measuring-rod and
the oscillation period of a clock if we move the measuring-rod and the clock, we turn
to the sine-Gordon equation. If however something happens, if the length and the
period of an oscillation during a motion really should change, then we would have to
re-think our above naive conception of measurement. In our case, we do not however
have to search for the answer to this question as Einstein had to, using brave deductions
out of a large theoretical principle. These deductions and the statements won from
them would have at that time to be proven by extensive precision measurements.
This is not necessary in our case. We can calculate the answer using the sineGordon equation. Here, we wish to express that the physical reliability of our calculations are founded on the Newtonian equations. These equations also control,
according to our Eq. (79), the dynamics of dislocations in crystals.
The starting point of our considerations is the inertial system, we call this the
reference system o , in which we have introduced, with the help of the line form q
I
o
a standard of length, a measuring-rod L o and with the help of an oscillating line q
III
o
a unit of measure for time, an oscillation period T o , in other words a clock. We thus
write
93
then measure the time of flight, as we defined above, of an object between A and
B. This is not the problem. As long as we as an observer stay in one and the same
inertial system with our crystal, no complications should arise.
However, what happens if we sit on the moving object and observe the crystal, if
we then wish to synchronise clocks and use our measuring-rod from our position on
the moving object? What happens if we compare the synchronised clocks and the
measuring-rods on the moving object with those of the observer at rest with respect
to the crystal? See Fig. 33. What velocity does the observer on the flying object use to
synchronise his clocks? Does he too use the sound velocity? But the sound velocity
changes if the observer moves with respect to the crystal, and it even becomes dependent upon the direction of motion! There are even several sound velocities inside of a
crystal. Which of these should we use? We will not found our further considerations
on sound velocity, the critical velocity for a propagation of an elastic deformation. In
1905, this was with the discovery of the Special Theory of Relativity by A. Einstein
completely different. In those days, the observer independent, untouchable, infallible
and unbelievable constancy of the speed of light in a vacuum c L , which represents
the signal velocity of an arbitrary electromagnetic excitation, was made to the infallible principle of the resulting theory, out of which everything else was consequently
deduced, cf. Einstein’s axioms quoted in Chap. 2, cf. also Chap. 16.
We do not have such a distinguished maximum velocity inside of a crystal—at
least not yet. We do however have something else. We know far more about our
instruments of measurement, our measuring-rods and clocks. They are solutions of
a physical equation, the sine-Gordon equation. For the mysterious question in the
Special Theory of Relativity, what happens with the length of the measuring-rod and
the oscillation period of a clock if we move the measuring-rod and the clock, we turn
to the sine-Gordon equation. If however something happens, if the length and the
period of an oscillation during a motion really should change, then we would have to
re-think our above naive conception of measurement. In our case, we do not however
have to search for the answer to this question as Einstein had to, using brave deductions
out of a large theoretical principle. These deductions and the statements won from
them would have at that time to be proven by extensive precision measurements.
This is not necessary in our case. We can calculate the answer using the sineGordon equation. Here, we wish to express that the physical reliability of our calculations are founded on the Newtonian equations. These equations also control,
according to our Eq. (79), the dynamics of dislocations in crystals.
The starting point of our considerations is the inertial system, we call this the
reference system o , in which we have introduced, with the help of the line form q
I
o
a standard of length, a measuring-rod L o and with the help of an oscillating line q
III
o
a unit of measure for time, an oscillation period T o , in other words a clock. We thus
write