12 The Measurement of the Critical Velocity
127
immediately form the hand setting of the synchronised clock U
∗
v in
. The arrival of
the signal at the clock U
∗
v will be called event C. We will therefore determine the hand
setting t
∗ of the clock U
∗
v when event C takes place. From o , we can determine
that the hand of U
∗
v for x = x and t = 0 is, according to (139), at the position t
B , as
was calculated in our example, see Fig. 12.6. Calculated from
, the signal does not
move yet, because t
B has a negative value and the signal only starts to move at t o
= 0.
Once again, observed from o , the travel time of the signal from the beginning to
the end point of the measuring section has, according to (131), the coefficient of
measure t → =
x
c o −v
. Because of time dilatation, the hand of the moving clock U
∗
v
only moves forward during this o time by t
→ = t →
1 − v 2 /c 2
o , so that at the
end it is positioned at t
∗ according to t
∗ = t
B + t
→ , thus with (139),
t
∗ =
−x v/c
2
o
1 − v 2 /c 2
o
+ t →
1 −
v 2
c 2
o
.
(145)
Hence, the event C, the arrival of the signal at U
∗
v is described by both reference
systems as follows,
: C
x
2 = x
, t
C = t
∗ =
−x v/c
2
o
1 − v 2 /c 2
o
+ t →
1 −
v 2
c 2
o
, (146)
o : C (x 2 = x , t C = t → ) .
(147)
The signals arrival at the clock U
∗
v has been shown in Fig. 12.9. With (113a) and
(145) we determine the velocity c o
in
,
c o
=
x
t
∗
=
x/
1 − v 2 /c 2
o
t →
1 − v 2 /c 2
o +
−x · v/c
2
o
1 − v 2 /c 2
o
=
x
t → (1 −
v
2
c 2
o
) −
x · v
c 2
o
,
and, if we take into account x//t → = c o − v according to (131) we get
c o
=
c o − v
1 − v 2 /c 2
o − (c o − v) v/c 2
o
= c
2
o
c o − v
c 2
o − v 2 − c o v + v 2 = c
2
o
c o − v
c o (c o − v)
,
and thus, see Günther [30],
c o
= c o .
(148)
This is in fact a remarkable result which contradicts everyday knowledge concerning
velocities. If the critical velocity c o is isotropic in just one reference system—we
assumed isotropy in the preferred frame o —then it is also isotropic in all other
reference systems
moving with an arbitrary constant velocity v with respect to
o , because of the behaviour of the measuring-rods and clocks. We will observe one
127
immediately form the hand setting of the synchronised clock U
∗
v in
. The arrival of
the signal at the clock U
∗
v will be called event C. We will therefore determine the hand
setting t
∗ of the clock U
∗
v when event C takes place. From o , we can determine
that the hand of U
∗
v for x = x and t = 0 is, according to (139), at the position t
B , as
was calculated in our example, see Fig. 12.6. Calculated from
, the signal does not
move yet, because t
B has a negative value and the signal only starts to move at t o
= 0.
Once again, observed from o , the travel time of the signal from the beginning to
the end point of the measuring section has, according to (131), the coefficient of
measure t → =
x
c o −v
. Because of time dilatation, the hand of the moving clock U
∗
v
only moves forward during this o time by t
→ = t →
1 − v 2 /c 2
o , so that at the
end it is positioned at t
∗ according to t
∗ = t
B + t
→ , thus with (139),
t
∗ =
−x v/c
2
o
1 − v 2 /c 2
o
+ t →
1 −
v 2
c 2
o
.
(145)
Hence, the event C, the arrival of the signal at U
∗
v is described by both reference
systems as follows,
: C
x
2 = x
, t
C = t
∗ =
−x v/c
2
o
1 − v 2 /c 2
o
+ t →
1 −
v 2
c 2
o
, (146)
o : C (x 2 = x , t C = t → ) .
(147)
The signals arrival at the clock U
∗
v has been shown in Fig. 12.9. With (113a) and
(145) we determine the velocity c o
in
,
c o
=
x
t
∗
=
x/
1 − v 2 /c 2
o
t →
1 − v 2 /c 2
o +
−x · v/c
2
o
1 − v 2 /c 2
o
=
x
t → (1 −
v
2
c 2
o
) −
x · v
c 2
o
,
and, if we take into account x//t → = c o − v according to (131) we get
c o
=
c o − v
1 − v 2 /c 2
o − (c o − v) v/c 2
o
= c
2
o
c o − v
c 2
o − v 2 − c o v + v 2 = c
2
o
c o − v
c o (c o − v)
,
and thus, see Günther [30],
c o
= c o .
(148)
This is in fact a remarkable result which contradicts everyday knowledge concerning
velocities. If the critical velocity c o is isotropic in just one reference system—we
assumed isotropy in the preferred frame o —then it is also isotropic in all other
reference systems
moving with an arbitrary constant velocity v with respect to
o , because of the behaviour of the measuring-rods and clocks. We will observe one