12 The Measurement of the Critical Velocity
115
Fig. 12.2 The determination of the average velocity ¯
c o . Here, it is assumed that the section to be
measured is moving with the velocity v = 0, 8 c o . This results in a factor
1 − v 2 /c 2
o = 0, 6. Thus,
L = 0, 6 · L o . If we take in comparison the coefficient of measure of time represented in Fig. 12.1,
t o = 2x/c o = 15, we calculate using (131) and (131a) due to the fact that we use the same x,
t → = t o
co
2(co−v) = 37, 5. This is shown on the right-hand static clock of o . On the other hand,
we get t ← = t o
co
2(co+v) = 4, 2. The total signal running time, the sum of both times, is shown on
the left-hand clock of o as the coefficient of measure t = t → + t ← = 42. In comparison, the
hand setting on the clock resting in is according to time dilatation t = t γ = 25. The dotted
line combines two points that describe one and the same event. We remind ourselves: An event is
defined by the position where it takes place and by the time when it takes place. The characterisation
of an event is dependent on the reference system in which it is described, by the statement of its
spacetime coordinates
the return journey, where the measuring section moves towards the signal with the
velocity v result in the effective velocity ¯
c o in (132), see Fig. 12.2, taking the mean,
1
¯
c o
=
1
2
1
c →
+
1
c ←
=
1
c o (1 − v 2 /c 2
o )
.
(132a)
We now consider the observer moving with the velocity v, who is thus in the reference
system
in which our measuring section X is at rest, which we can illustrate to
ourselves as a resting ‘rod’ in
. The decisive question is: What value c o
does the
observer in his reference system
find out for the velocity with which our signal
propagates? How large is the, from the preferred frame o , c o attributed velocity for
the observer moving with the velocity v relative to the crystal lattice?
We will answer these questions in two steps. We will first ask, what value ¯
c o
does
the observer discover in
for the effective velocity that the signal needs to traverse
back and forth along the measuring section X ?
We can answer this question without any further assumptions. If the observer wants to
measure this velocity ¯
c o
, all he has to do is to copy our method where we determined
the effective velocity ¯
c o , cf. (132). The only difference is that he uses the measuringrods and clocks which are at rest relative to him. Let us observe how he proceeds.
115
Fig. 12.2 The determination of the average velocity ¯
c o . Here, it is assumed that the section to be
measured is moving with the velocity v = 0, 8 c o . This results in a factor
1 − v 2 /c 2
o = 0, 6. Thus,
L = 0, 6 · L o . If we take in comparison the coefficient of measure of time represented in Fig. 12.1,
t o = 2x/c o = 15, we calculate using (131) and (131a) due to the fact that we use the same x,
t → = t o
co
2(co−v) = 37, 5. This is shown on the right-hand static clock of o . On the other hand,
we get t ← = t o
co
2(co+v) = 4, 2. The total signal running time, the sum of both times, is shown on
the left-hand clock of o as the coefficient of measure t = t → + t ← = 42. In comparison, the
hand setting on the clock resting in is according to time dilatation t = t γ = 25. The dotted
line combines two points that describe one and the same event. We remind ourselves: An event is
defined by the position where it takes place and by the time when it takes place. The characterisation
of an event is dependent on the reference system in which it is described, by the statement of its
spacetime coordinates
the return journey, where the measuring section moves towards the signal with the
velocity v result in the effective velocity ¯
c o in (132), see Fig. 12.2, taking the mean,
1
¯
c o
=
1
2
1
c →
+
1
c ←
=
1
c o (1 − v 2 /c 2
o )
.
(132a)
We now consider the observer moving with the velocity v, who is thus in the reference
system
in which our measuring section X is at rest, which we can illustrate to
ourselves as a resting ‘rod’ in
. The decisive question is: What value c o
does the
observer in his reference system
find out for the velocity with which our signal
propagates? How large is the, from the preferred frame o , c o attributed velocity for
the observer moving with the velocity v relative to the crystal lattice?
We will answer these questions in two steps. We will first ask, what value ¯
c o
does
the observer discover in
for the effective velocity that the signal needs to traverse
back and forth along the measuring section X ?
We can answer this question without any further assumptions. If the observer wants to
measure this velocity ¯
c o
, all he has to do is to copy our method where we determined
the effective velocity ¯
c o , cf. (132). The only difference is that he uses the measuringrods and clocks which are at rest relative to him. Let us observe how he proceeds.