116
12 The Measurement of the Critical Velocity
His measuring-rod L
is according to Eq. (112) contracted in comparison with our
measuring rod L o by the factor
1 − v 2 /c 2
o . Thus, the coefficient of measure x
that
he determines for the rod X resting with respect to him is larger than our coefficient
of measure x by this factor, in other words x
= x/
1 − v 2 /c 2
o , just as we have
written it above in Eq. (113a), see also Fig. 12.2. Besides this, the observers clock
positioned at the starting point of the measuring section and moved (in reference
to o ) with the velocity v together with this measuring section shows for the time
interval between the event of sound signals emission and the event of its arrival the
coefficient of measure t
that has to be calculated using Eq. (121a) for time dilatation
of moving clocks. The coefficient of measure t
is smaller than the measured (with
the help of two clocks) coefficient of measure in o , t = t → + t ← by the factor
1 − v 2 /c 2
o for the difference in time of both events, see Fig. 12.2
t
= t
1 −
v 2
c 2
o
.
(121a)
The moving observer in
thus measures according to (113a) and (121a) a velocity
¯
c o
according to
¯
c o
=
2x
t =
2x
t
1
1 − v 2 /c 2
o
.
We have calculated the factor 2x//t = ¯
c o in (132). Hence, the observer moving
with respect to the crystal lattice with the velocity v measures for the effective velocity
¯
c o
of the signal in his reference system
the value
¯
c o
= c o
1 − v
2
/c
2
o
1 − v 2 /c 2
o
,
and thus, as we have shown in 1994, cf. G¨ unther [34],
¯
c o
= c o .
(133)
As remarkable as this conformity between both velocities may seem, the quantity
¯
c o
measured by the observer in
is an effective velocity. If we denote the above
velocity of the signal on the first part of the way as c
→ and on the way back as c
←
we can calculate ¯
c o
as shown above by taking the mean,
1
¯
c o
=
1
2
1
c
→
+
1
c
←
=
1
c o
.
(133a)
Our opinion however, which should be considered cautiously, tends towards the
general conditions depicted in our preferred frame o , which leads to the view that
the velocity which is effective to traverse the measuring section on the way back
is greater than on the way there. However, if we could be certain that the velocity
12 The Measurement of the Critical Velocity
His measuring-rod L
is according to Eq. (112) contracted in comparison with our
measuring rod L o by the factor
1 − v 2 /c 2
o . Thus, the coefficient of measure x
that
he determines for the rod X resting with respect to him is larger than our coefficient
of measure x by this factor, in other words x
= x/
1 − v 2 /c 2
o , just as we have
written it above in Eq. (113a), see also Fig. 12.2. Besides this, the observers clock
positioned at the starting point of the measuring section and moved (in reference
to o ) with the velocity v together with this measuring section shows for the time
interval between the event of sound signals emission and the event of its arrival the
coefficient of measure t
that has to be calculated using Eq. (121a) for time dilatation
of moving clocks. The coefficient of measure t
is smaller than the measured (with
the help of two clocks) coefficient of measure in o , t = t → + t ← by the factor
1 − v 2 /c 2
o for the difference in time of both events, see Fig. 12.2
t
= t
1 −
v 2
c 2
o
.
(121a)
The moving observer in
thus measures according to (113a) and (121a) a velocity
¯
c o
according to
¯
c o
=
2x
t =
2x
t
1
1 − v 2 /c 2
o
.
We have calculated the factor 2x//t = ¯
c o in (132). Hence, the observer moving
with respect to the crystal lattice with the velocity v measures for the effective velocity
¯
c o
of the signal in his reference system
the value
¯
c o
= c o
1 − v
2
/c
2
o
1 − v 2 /c 2
o
,
and thus, as we have shown in 1994, cf. G¨ unther [34],
¯
c o
= c o .
(133)
As remarkable as this conformity between both velocities may seem, the quantity
¯
c o
measured by the observer in
is an effective velocity. If we denote the above
velocity of the signal on the first part of the way as c
→ and on the way back as c
←
we can calculate ¯
c o
as shown above by taking the mean,
1
¯
c o
=
1
2
1
c
→
+
1
c
←
=
1
c o
.
(133a)
Our opinion however, which should be considered cautiously, tends towards the
general conditions depicted in our preferred frame o , which leads to the view that
the velocity which is effective to traverse the measuring section on the way back
is greater than on the way there. However, if we could be certain that the velocity